{"id":65461,"date":"2026-08-17T16:21:10","date_gmt":"2026-08-17T15:21:10","guid":{"rendered":"https:\/\/blog-admin.thethinkacademy.com\/?p=65461"},"modified":"2026-08-17T16:21:12","modified_gmt":"2026-08-17T15:21:12","slug":"cevas-theorem-guide","status":"publish","type":"post","link":"https:\/\/blog-admin.thethinkacademy.com\/blog\/2026\/08\/17\/cevas-theorem-guide\/","title":{"rendered":"Ceva&#8217;s Theorem Explained: Definition, Proof and Applications"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Ceva&#8217;s theorem is one of the most elegant and useful results in triangle geometry. It gives a precise, elegant condition for when three lines drawn from the vertices of a triangle all pass through a single point \u2014 a property called <strong>concurrency<\/strong>. It is a standard tool in competition geometry at the AMC 10, AMC 12, Euclid, and COMC level, and it connects to some of the most beautiful classical results in Euclidean geometry.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">What Is a Cevian?<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Before stating Ceva&#8217;s theorem, the key term: a <strong>cevian<\/strong> of a triangle is a line segment from a vertex of the triangle to a point on the opposite side (or its extension).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In triangle <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><mi>B<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">ABC<\/annotation><\/semantics><\/math>ABC:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>A cevian from vertex <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">A<\/annotation><\/semantics><\/math>A meets the opposite side <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>B<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">BC<\/annotation><\/semantics><\/math>BC at some point <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">D<\/annotation><\/semantics><\/math>D<\/li>\n\n\n\n<li>A cevian from vertex <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>B<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">B<\/annotation><\/semantics><\/math>B meets the opposite side <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>C<\/mi><mi>A<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">CA<\/annotation><\/semantics><\/math>CA at some point <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>E<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">E<\/annotation><\/semantics><\/math>E<\/li>\n\n\n\n<li>A cevian from vertex <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">C<\/annotation><\/semantics><\/math>C meets the opposite side <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><mi>B<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">AB<\/annotation><\/semantics><\/math>AB at some point <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>F<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">F<\/annotation><\/semantics><\/math>F<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Famous cevians:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The <strong>medians<\/strong> \u2014 cevians from each vertex to the midpoint of the opposite side. All three medians are concurrent at the <strong>centroid<\/strong>.<\/li>\n\n\n\n<li>The <strong>altitudes<\/strong> \u2014 perpendiculars from each vertex to the opposite side. All three altitudes are concurrent at the <strong>orthocentre<\/strong>.<\/li>\n\n\n\n<li>The <strong>angle bisectors<\/strong> \u2014 bisectors of each interior angle. All three are concurrent at the <strong>incentre<\/strong>.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">These three classical facts \u2014 medians, altitudes, and angle bisectors each being concurrent \u2014 are all special cases of Ceva&#8217;s theorem.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Ceva&#8217;s Theorem: Statement<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Ceva&#8217;s Theorem:<\/strong> In triangle <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><mi>B<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">ABC<\/annotation><\/semantics><\/math>ABC, let <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">D<\/annotation><\/semantics><\/math>D, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>E<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">E<\/annotation><\/semantics><\/math>E, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>F<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">F<\/annotation><\/semantics><\/math>F be points on sides <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>B<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">BC<\/annotation><\/semantics><\/math>BC, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>C<\/mi><mi>A<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">CA<\/annotation><\/semantics><\/math>CA, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><mi>B<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">AB<\/annotation><\/semantics><\/math>AB respectively (or their extensions). The cevians <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><mi>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">AD<\/annotation><\/semantics><\/math>AD, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>B<\/mi><mi>E<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">BE<\/annotation><\/semantics><\/math>BE, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>C<\/mi><mi>F<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">CF<\/annotation><\/semantics><\/math>CF are concurrent (all pass through a single point) <strong>if and only if<\/strong>:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mrow><mi>B<\/mi><mi>D<\/mi><\/mrow><mrow><mi>D<\/mi><mi>C<\/mi><\/mrow><\/mfrac><mo>\u22c5<\/mo><mfrac><mrow><mi>C<\/mi><mi>E<\/mi><\/mrow><mrow><mi>E<\/mi><mi>A<\/mi><\/mrow><\/mfrac><mo>\u22c5<\/mo><mfrac><mrow><mi>A<\/mi><mi>F<\/mi><\/mrow><mrow><mi>F<\/mi><mi>B<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{BD}{DC} \\cdot \\frac{CE}{EA} \\cdot \\frac{AF}{FB} = 1<\/annotation><\/semantics><\/math>DCBD\u200b\u22c5EACE\u200b\u22c5FBAF\u200b=1<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Note on signs:<\/strong> When working with points on the extensions of sides (outside the triangle), signed ratios are used. For points inside the triangle, all ratios are positive and the product equals 1 with all positive values. The signed version (sometimes called the <strong>trigonometric form<\/strong> when combined with angles) generalises to points outside the triangle.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>The &#8220;if and only if&#8221;:<\/strong> Ceva&#8217;s theorem works in both directions:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>If the three cevians are concurrent, the ratio product equals 1.<\/li>\n\n\n\n<li>If the ratio product equals 1, the three cevians are concurrent.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">This makes it both a test for concurrency and a tool for constructing concurrent cevians.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Proof of Ceva&#8217;s Theorem<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">The most accessible proof uses areas.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Setup:<\/strong> Let the three cevians <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><mi>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">AD<\/annotation><\/semantics><\/math>AD, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>B<\/mi><mi>E<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">BE<\/annotation><\/semantics><\/math>BE, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>C<\/mi><mi>F<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">CF<\/annotation><\/semantics><\/math>CF be concurrent at point <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>P<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">P<\/annotation><\/semantics><\/math>P inside triangle <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><mi>B<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">ABC<\/annotation><\/semantics><\/math>ABC.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Step 1:<\/strong> Express <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>B<\/mi><mi>D<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>D<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">BD\/DC<\/annotation><\/semantics><\/math>BD\/DC using areas.<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mrow><mi>B<\/mi><mi>D<\/mi><\/mrow><mrow><mi>D<\/mi><mi>C<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mo stretchy=\"false\">[<\/mo><mi mathvariant=\"normal\">\u25b3<\/mi><mi>A<\/mi><mi>B<\/mi><mi>D<\/mi><mo stretchy=\"false\">]<\/mo><\/mrow><mrow><mo stretchy=\"false\">[<\/mo><mi mathvariant=\"normal\">\u25b3<\/mi><mi>A<\/mi><mi>C<\/mi><mi>D<\/mi><mo stretchy=\"false\">]<\/mo><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mo stretchy=\"false\">[<\/mo><mi mathvariant=\"normal\">\u25b3<\/mi><mi>P<\/mi><mi>B<\/mi><mi>D<\/mi><mo stretchy=\"false\">]<\/mo><\/mrow><mrow><mo stretchy=\"false\">[<\/mo><mi mathvariant=\"normal\">\u25b3<\/mi><mi>P<\/mi><mi>C<\/mi><mi>D<\/mi><mo stretchy=\"false\">]<\/mo><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{BD}{DC} = \\frac{[\\triangle ABD]}{[\\triangle ACD]} = \\frac{[\\triangle PBD]}{[\\triangle PCD]}<\/annotation><\/semantics><\/math>DCBD\u200b=[\u25b3ACD][\u25b3ABD]\u200b=[\u25b3PCD][\u25b3PBD]\u200b<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">where <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo stretchy=\"false\">[<\/mo><mi mathvariant=\"normal\">\u25b3<\/mi><mi>X<\/mi><mi>Y<\/mi><mi>Z<\/mi><mo stretchy=\"false\">]<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">[\\triangle XYZ]<\/annotation><\/semantics><\/math>[\u25b3XYZ] denotes the area of triangle <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>X<\/mi><mi>Y<\/mi><mi>Z<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">XYZ<\/annotation><\/semantics><\/math>XYZ. Both ratios hold because triangles sharing the same height have areas proportional to their bases.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Combining these (using the property <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mfrac><mi>a<\/mi><mi>b<\/mi><\/mfrac><mo>=<\/mo><mfrac><mi>c<\/mi><mi>d<\/mi><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>a<\/mi><mo>+<\/mo><mi>c<\/mi><\/mrow><mrow><mi>b<\/mi><mo>+<\/mo><mi>d<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{a}{b} = \\frac{c}{d} = \\frac{a+c}{b+d}<\/annotation><\/semantics><\/math>ba\u200b=dc\u200b=b+da+c\u200b):<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mrow><mi>B<\/mi><mi>D<\/mi><\/mrow><mrow><mi>D<\/mi><mi>C<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mo stretchy=\"false\">[<\/mo><mi mathvariant=\"normal\">\u25b3<\/mi><mi>A<\/mi><mi>B<\/mi><mi>D<\/mi><mo stretchy=\"false\">]<\/mo><mo>+<\/mo><mo stretchy=\"false\">[<\/mo><mi mathvariant=\"normal\">\u25b3<\/mi><mi>P<\/mi><mi>B<\/mi><mi>D<\/mi><mo stretchy=\"false\">]<\/mo><\/mrow><mrow><mo stretchy=\"false\">[<\/mo><mi mathvariant=\"normal\">\u25b3<\/mi><mi>A<\/mi><mi>C<\/mi><mi>D<\/mi><mo stretchy=\"false\">]<\/mo><mo>+<\/mo><mo stretchy=\"false\">[<\/mo><mi mathvariant=\"normal\">\u25b3<\/mi><mi>P<\/mi><mi>C<\/mi><mi>D<\/mi><mo stretchy=\"false\">]<\/mo><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{BD}{DC} = \\frac{[\\triangle ABD] + [\\triangle PBD]}{[\\triangle ACD] + [\\triangle PCD]}<\/annotation><\/semantics><\/math>DCBD\u200b=[\u25b3ACD]+[\u25b3PCD][\u25b3ABD]+[\u25b3PBD]\u200b<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Wait \u2014 let&#8217;s use the cleaner form. Since <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">A<\/annotation><\/semantics><\/math>A, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>B<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">B<\/annotation><\/semantics><\/math>B, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">D<\/annotation><\/semantics><\/math>D lie with <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">D<\/annotation><\/semantics><\/math>D on <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>B<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">BC<\/annotation><\/semantics><\/math>BC:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mrow><mi>B<\/mi><mi>D<\/mi><\/mrow><mrow><mi>D<\/mi><mi>C<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mo stretchy=\"false\">[<\/mo><mi mathvariant=\"normal\">\u25b3<\/mi><mi>A<\/mi><mi>B<\/mi><mi>D<\/mi><mo stretchy=\"false\">]<\/mo><\/mrow><mrow><mo stretchy=\"false\">[<\/mo><mi mathvariant=\"normal\">\u25b3<\/mi><mi>A<\/mi><mi>C<\/mi><mi>D<\/mi><mo stretchy=\"false\">]<\/mo><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{BD}{DC} = \\frac{[\\triangle ABD]}{[\\triangle ACD]}<\/annotation><\/semantics><\/math>DCBD\u200b=[\u25b3ACD][\u25b3ABD]\u200b<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">and also:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mrow><mi>B<\/mi><mi>D<\/mi><\/mrow><mrow><mi>D<\/mi><mi>C<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mo stretchy=\"false\">[<\/mo><mi mathvariant=\"normal\">\u25b3<\/mi><mi>P<\/mi><mi>B<\/mi><mi>D<\/mi><mo stretchy=\"false\">]<\/mo><\/mrow><mrow><mo stretchy=\"false\">[<\/mo><mi mathvariant=\"normal\">\u25b3<\/mi><mi>P<\/mi><mi>C<\/mi><mi>D<\/mi><mo stretchy=\"false\">]<\/mo><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{BD}{DC} = \\frac{[\\triangle PBD]}{[\\triangle PCD]}<\/annotation><\/semantics><\/math>DCBD\u200b=[\u25b3PCD][\u25b3PBD]\u200b<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Using the ratio-sum property:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mrow><mi>B<\/mi><mi>D<\/mi><\/mrow><mrow><mi>D<\/mi><mi>C<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mo stretchy=\"false\">[<\/mo><mi mathvariant=\"normal\">\u25b3<\/mi><mi>A<\/mi><mi>B<\/mi><mi>P<\/mi><mo stretchy=\"false\">]<\/mo><\/mrow><mrow><mo stretchy=\"false\">[<\/mo><mi mathvariant=\"normal\">\u25b3<\/mi><mi>A<\/mi><mi>C<\/mi><mi>P<\/mi><mo stretchy=\"false\">]<\/mo><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{BD}{DC} = \\frac{[\\triangle ABP]}{[\\triangle ACP]}<\/annotation><\/semantics><\/math>DCBD\u200b=[\u25b3ACP][\u25b3ABP]\u200b<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Step 2:<\/strong> Similarly:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mrow><mi>C<\/mi><mi>E<\/mi><\/mrow><mrow><mi>E<\/mi><mi>A<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mo stretchy=\"false\">[<\/mo><mi mathvariant=\"normal\">\u25b3<\/mi><mi>B<\/mi><mi>C<\/mi><mi>P<\/mi><mo stretchy=\"false\">]<\/mo><\/mrow><mrow><mo stretchy=\"false\">[<\/mo><mi mathvariant=\"normal\">\u25b3<\/mi><mi>A<\/mi><mi>B<\/mi><mi>P<\/mi><mo stretchy=\"false\">]<\/mo><\/mrow><\/mfrac><mo separator=\"true\">,<\/mo><mspace width=\"1em\"><\/mspace><mfrac><mrow><mi>A<\/mi><mi>F<\/mi><\/mrow><mrow><mi>F<\/mi><mi>B<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mo stretchy=\"false\">[<\/mo><mi mathvariant=\"normal\">\u25b3<\/mi><mi>A<\/mi><mi>C<\/mi><mi>P<\/mi><mo stretchy=\"false\">]<\/mo><\/mrow><mrow><mo stretchy=\"false\">[<\/mo><mi mathvariant=\"normal\">\u25b3<\/mi><mi>B<\/mi><mi>C<\/mi><mi>P<\/mi><mo stretchy=\"false\">]<\/mo><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{CE}{EA} = \\frac{[\\triangle BCP]}{[\\triangle ABP]}, \\quad \\frac{AF}{FB} = \\frac{[\\triangle ACP]}{[\\triangle BCP]}<\/annotation><\/semantics><\/math>EACE\u200b=[\u25b3ABP][\u25b3BCP]\u200b,FBAF\u200b=[\u25b3BCP][\u25b3ACP]\u200b<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Step 3:<\/strong> Multiply:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mrow><mi>B<\/mi><mi>D<\/mi><\/mrow><mrow><mi>D<\/mi><mi>C<\/mi><\/mrow><\/mfrac><mo>\u22c5<\/mo><mfrac><mrow><mi>C<\/mi><mi>E<\/mi><\/mrow><mrow><mi>E<\/mi><mi>A<\/mi><\/mrow><\/mfrac><mo>\u22c5<\/mo><mfrac><mrow><mi>A<\/mi><mi>F<\/mi><\/mrow><mrow><mi>F<\/mi><mi>B<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mo stretchy=\"false\">[<\/mo><mi mathvariant=\"normal\">\u25b3<\/mi><mi>A<\/mi><mi>B<\/mi><mi>P<\/mi><mo stretchy=\"false\">]<\/mo><\/mrow><mrow><mo stretchy=\"false\">[<\/mo><mi mathvariant=\"normal\">\u25b3<\/mi><mi>A<\/mi><mi>C<\/mi><mi>P<\/mi><mo stretchy=\"false\">]<\/mo><\/mrow><\/mfrac><mo>\u22c5<\/mo><mfrac><mrow><mo stretchy=\"false\">[<\/mo><mi mathvariant=\"normal\">\u25b3<\/mi><mi>B<\/mi><mi>C<\/mi><mi>P<\/mi><mo stretchy=\"false\">]<\/mo><\/mrow><mrow><mo stretchy=\"false\">[<\/mo><mi mathvariant=\"normal\">\u25b3<\/mi><mi>A<\/mi><mi>B<\/mi><mi>P<\/mi><mo stretchy=\"false\">]<\/mo><\/mrow><\/mfrac><mo>\u22c5<\/mo><mfrac><mrow><mo stretchy=\"false\">[<\/mo><mi mathvariant=\"normal\">\u25b3<\/mi><mi>A<\/mi><mi>C<\/mi><mi>P<\/mi><mo stretchy=\"false\">]<\/mo><\/mrow><mrow><mo stretchy=\"false\">[<\/mo><mi mathvariant=\"normal\">\u25b3<\/mi><mi>B<\/mi><mi>C<\/mi><mi>P<\/mi><mo stretchy=\"false\">]<\/mo><\/mrow><\/mfrac><mo>=<\/mo><mn>1<\/mn><mspace width=\"1em\"><\/mspace><mi mathvariant=\"normal\">\u25a1<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{BD}{DC} \\cdot \\frac{CE}{EA} \\cdot \\frac{AF}{FB} = \\frac{[\\triangle ABP]}{[\\triangle ACP]} \\cdot \\frac{[\\triangle BCP]}{[\\triangle ABP]} \\cdot \\frac{[\\triangle ACP]}{[\\triangle BCP]} = 1 \\quad \\square<\/annotation><\/semantics><\/math>DCBD\u200b\u22c5EACE\u200b\u22c5FBAF\u200b=[\u25b3ACP][\u25b3ABP]\u200b\u22c5[\u25b3ABP][\u25b3BCP]\u200b\u22c5[\u25b3BCP][\u25b3ACP]\u200b=1\u25a1<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>The converse:<\/strong> If the product equals 1, let <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>C<\/mi><msup><mi>F<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">CF&#8217;<\/annotation><\/semantics><\/math>CF\u2032 be the cevian through the intersection of <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><mi>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">AD<\/annotation><\/semantics><\/math>AD and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>B<\/mi><mi>E<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">BE<\/annotation><\/semantics><\/math>BE. Then by the forward direction, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mfrac><mrow><mi>A<\/mi><msup><mi>F<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><mrow><msup><mi>F<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mi>B<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>A<\/mi><mi>F<\/mi><\/mrow><mrow><mi>F<\/mi><mi>B<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{AF&#8217;}{F&#8217;B} = \\frac{AF}{FB}<\/annotation><\/semantics><\/math>F\u2032BAF\u2032\u200b=FBAF\u200b, so <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>F<\/mi><mo>=<\/mo><msup><mi>F<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">F = F&#8217;<\/annotation><\/semantics><\/math>F=F\u2032 and the three cevians are concurrent. \u25a1<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">The Three Classical Concurrencies: Ceva&#8217;s Theorem in Action<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">The Centroid (Medians are Concurrent)<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">The medians connect each vertex to the midpoint of the opposite side:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">D<\/annotation><\/semantics><\/math>D = midpoint of <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>B<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">BC<\/annotation><\/semantics><\/math>BC: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>B<\/mi><mi>D<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>D<\/mi><mi>C<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">BD\/DC = 1<\/annotation><\/semantics><\/math>BD\/DC=1 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>E<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">E<\/annotation><\/semantics><\/math>E = midpoint of <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>C<\/mi><mi>A<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">CA<\/annotation><\/semantics><\/math>CA: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>C<\/mi><mi>E<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>E<\/mi><mi>A<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">CE\/EA = 1<\/annotation><\/semantics><\/math>CE\/EA=1 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>F<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">F<\/annotation><\/semantics><\/math>F = midpoint of <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><mi>B<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">AB<\/annotation><\/semantics><\/math>AB: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><mi>F<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>F<\/mi><mi>B<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">AF\/FB = 1<\/annotation><\/semantics><\/math>AF\/FB=1<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Product: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>1<\/mn><mo>\u00d7<\/mo><mn>1<\/mn><mo>\u00d7<\/mo><mn>1<\/mn><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">1 \\times 1 \\times 1 = 1<\/annotation><\/semantics><\/math>1\u00d71\u00d71=1 \u2713<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Ceva&#8217;s theorem confirms the medians are concurrent.<\/strong> The point of concurrence is the centroid, which divides each median in the ratio 2:1 from the vertex.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">The Angle Bisector Concurrency (Incentre)<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">By the <strong>Angle Bisector Theorem<\/strong>: the bisector from vertex <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">A<\/annotation><\/semantics><\/math>A to side <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>B<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">BC<\/annotation><\/semantics><\/math>BC divides <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>B<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">BC<\/annotation><\/semantics><\/math>BC in the ratio $AB:AC = c:b$.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">So: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>B<\/mi><mi>D<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>D<\/mi><mi>C<\/mi><mo>=<\/mo><mi>c<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>b<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">BD\/DC = c\/b<\/annotation><\/semantics><\/math>BD\/DC=c\/b, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>C<\/mi><mi>E<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>E<\/mi><mi>A<\/mi><mo>=<\/mo><mi>a<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>c<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">CE\/EA = a\/c<\/annotation><\/semantics><\/math>CE\/EA=a\/c, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><mi>F<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>F<\/mi><mi>B<\/mi><mo>=<\/mo><mi>b<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">AF\/FB = b\/a<\/annotation><\/semantics><\/math>AF\/FB=b\/a<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Product: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mfrac><mi>c<\/mi><mi>b<\/mi><\/mfrac><mo>\u22c5<\/mo><mfrac><mi>a<\/mi><mi>c<\/mi><\/mfrac><mo>\u22c5<\/mo><mfrac><mi>b<\/mi><mi>a<\/mi><\/mfrac><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{c}{b} \\cdot \\frac{a}{c} \\cdot \\frac{b}{a} = 1<\/annotation><\/semantics><\/math>bc\u200b\u22c5ca\u200b\u22c5ab\u200b=1 \u2713<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>The angle bisectors are concurrent<\/strong> at the incentre.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">The Altitude Concurrency (Orthocentre)<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">In triangle <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><mi>B<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">ABC<\/annotation><\/semantics><\/math>ABC with altitude foot <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">D<\/annotation><\/semantics><\/math>D on <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>B<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">BC<\/annotation><\/semantics><\/math>BC from <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">A<\/annotation><\/semantics><\/math>A:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Using trigonometry: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>B<\/mi><mi>D<\/mi><mo>=<\/mo><mi>c<\/mi><mi>cos<\/mi><mo>\u2061<\/mo><mi>B<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">BD = c\\cos B<\/annotation><\/semantics><\/math>BD=ccosB and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>D<\/mi><mi>C<\/mi><mo>=<\/mo><mi>b<\/mi><mi>cos<\/mi><mo>\u2061<\/mo><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">DC = b\\cos C<\/annotation><\/semantics><\/math>DC=bcosC, so <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>B<\/mi><mi>D<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>D<\/mi><mi>C<\/mi><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><mi>c<\/mi><mi>cos<\/mi><mo>\u2061<\/mo><mi>B<\/mi><mo stretchy=\"false\">)<\/mo><mi mathvariant=\"normal\">\/<\/mi><mo stretchy=\"false\">(<\/mo><mi>b<\/mi><mi>cos<\/mi><mo>\u2061<\/mo><mi>C<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">BD\/DC = (c\\cos B)\/(b\\cos C)<\/annotation><\/semantics><\/math>BD\/DC=(ccosB)\/(bcosC).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Similarly: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>C<\/mi><mi>E<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>E<\/mi><mi>A<\/mi><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mi>cos<\/mi><mo>\u2061<\/mo><mi>C<\/mi><mo stretchy=\"false\">)<\/mo><mi mathvariant=\"normal\">\/<\/mi><mo stretchy=\"false\">(<\/mo><mi>c<\/mi><mi>cos<\/mi><mo>\u2061<\/mo><mi>A<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">CE\/EA = (a\\cos C)\/(c\\cos A)<\/annotation><\/semantics><\/math>CE\/EA=(acosC)\/(ccosA) and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><mi>F<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>F<\/mi><mi>B<\/mi><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><mi>b<\/mi><mi>cos<\/mi><mo>\u2061<\/mo><mi>A<\/mi><mo stretchy=\"false\">)<\/mo><mi mathvariant=\"normal\">\/<\/mi><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mi>cos<\/mi><mo>\u2061<\/mo><mi>B<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">AF\/FB = (b\\cos A)\/(a\\cos B)<\/annotation><\/semantics><\/math>AF\/FB=(bcosA)\/(acosB).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Product: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mfrac><mrow><mi>c<\/mi><mi>cos<\/mi><mo>\u2061<\/mo><mi>B<\/mi><\/mrow><mrow><mi>b<\/mi><mi>cos<\/mi><mo>\u2061<\/mo><mi>C<\/mi><\/mrow><\/mfrac><mo>\u22c5<\/mo><mfrac><mrow><mi>a<\/mi><mi>cos<\/mi><mo>\u2061<\/mo><mi>C<\/mi><\/mrow><mrow><mi>c<\/mi><mi>cos<\/mi><mo>\u2061<\/mo><mi>A<\/mi><\/mrow><\/mfrac><mo>\u22c5<\/mo><mfrac><mrow><mi>b<\/mi><mi>cos<\/mi><mo>\u2061<\/mo><mi>A<\/mi><\/mrow><mrow><mi>a<\/mi><mi>cos<\/mi><mo>\u2061<\/mo><mi>B<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{c\\cos B}{b\\cos C} \\cdot \\frac{a\\cos C}{c\\cos A} \\cdot \\frac{b\\cos A}{a\\cos B} = 1<\/annotation><\/semantics><\/math>bcosCccosB\u200b\u22c5ccosAacosC\u200b\u22c5acosBbcosA\u200b=1 \u2713<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>The altitudes are concurrent<\/strong> at the orthocentre.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Three classical theorems \u2014 all special cases of Ceva&#8217;s theorem \u2014 proved in a few lines each.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><a href=\"https:\/\/www.thinkacademy.ca\/free-assessment?source_id=6721&amp;source_type=9&amp;utm_medium=website&amp;utm_source=pc_blog\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"409\" src=\"https:\/\/blog-admin.thethinkacademy.com\/wp-content\/uploads\/2026\/08\/ChatGPT-Image-Aug-17-2026-04_02_28-PM-1-1024x409.png\" alt=\"\" class=\"wp-image-65463\" srcset=\"https:\/\/blog-admin.thethinkacademy.com\/wp-content\/uploads\/2026\/08\/ChatGPT-Image-Aug-17-2026-04_02_28-PM-1-1024x409.png 1024w, https:\/\/blog-admin.thethinkacademy.com\/wp-content\/uploads\/2026\/08\/ChatGPT-Image-Aug-17-2026-04_02_28-PM-1-300x120.png 300w, https:\/\/blog-admin.thethinkacademy.com\/wp-content\/uploads\/2026\/08\/ChatGPT-Image-Aug-17-2026-04_02_28-PM-1-768x307.png 768w, https:\/\/blog-admin.thethinkacademy.com\/wp-content\/uploads\/2026\/08\/ChatGPT-Image-Aug-17-2026-04_02_28-PM-1-1536x614.png 1536w, https:\/\/blog-admin.thethinkacademy.com\/wp-content\/uploads\/2026\/08\/ChatGPT-Image-Aug-17-2026-04_02_28-PM-1-1300x520.png 1300w, https:\/\/blog-admin.thethinkacademy.com\/wp-content\/uploads\/2026\/08\/ChatGPT-Image-Aug-17-2026-04_02_28-PM-1.png 1983w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/a><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Worked Examples<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">Example 1 \u2014 Testing Concurrency (AMC\/Competition Level)<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>In triangle ABCABC ABC, point DD D is on BCBC BC with $BD:DC = 2:3, point $E is on CACA CA with $CE:EA = 4:1, and point $F is on ABAB AB with $AF:FB = 3:8. Are the cevians $AD , BEBE BE, CFCF CF concurrent?<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Compute the product:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mrow><mi>B<\/mi><mi>D<\/mi><\/mrow><mrow><mi>D<\/mi><mi>C<\/mi><\/mrow><\/mfrac><mo>\u22c5<\/mo><mfrac><mrow><mi>C<\/mi><mi>E<\/mi><\/mrow><mrow><mi>E<\/mi><mi>A<\/mi><\/mrow><\/mfrac><mo>\u22c5<\/mo><mfrac><mrow><mi>A<\/mi><mi>F<\/mi><\/mrow><mrow><mi>F<\/mi><mi>B<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mn>2<\/mn><mn>3<\/mn><\/mfrac><mo>\u22c5<\/mo><mfrac><mn>4<\/mn><mn>1<\/mn><\/mfrac><mo>\u22c5<\/mo><mfrac><mn>3<\/mn><mn>8<\/mn><\/mfrac><mo>=<\/mo><mfrac><mrow><mn>2<\/mn><mo>\u00d7<\/mo><mn>4<\/mn><mo>\u00d7<\/mo><mn>3<\/mn><\/mrow><mrow><mn>3<\/mn><mo>\u00d7<\/mo><mn>1<\/mn><mo>\u00d7<\/mo><mn>8<\/mn><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mn>24<\/mn><mn>24<\/mn><\/mfrac><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{BD}{DC} \\cdot \\frac{CE}{EA} \\cdot \\frac{AF}{FB} = \\frac{2}{3} \\cdot \\frac{4}{1} \\cdot \\frac{3}{8} = \\frac{2 \\times 4 \\times 3}{3 \\times 1 \\times 8} = \\frac{24}{24} = 1<\/annotation><\/semantics><\/math>DCBD\u200b\u22c5EACE\u200b\u22c5FBAF\u200b=32\u200b\u22c514\u200b\u22c583\u200b=3\u00d71\u00d782\u00d74\u00d73\u200b=2424\u200b=1<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Yes \u2014 the cevians are concurrent.<\/strong> \u2713<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Example 2 \u2014 Finding an Unknown Ratio<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>In triangle ABCABC ABC, cevians ADAD AD, BEBE BE, CFCF CF are concurrent. Given BD\/DC=3BD\/DC = 3 BD\/DC=3, CE\/EA=2CE\/EA = 2 CE\/EA=2, find AF\/FBAF\/FB AF\/FB.<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">By Ceva&#8217;s theorem:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mrow><mi>B<\/mi><mi>D<\/mi><\/mrow><mrow><mi>D<\/mi><mi>C<\/mi><\/mrow><\/mfrac><mo>\u22c5<\/mo><mfrac><mrow><mi>C<\/mi><mi>E<\/mi><\/mrow><mrow><mi>E<\/mi><mi>A<\/mi><\/mrow><\/mfrac><mo>\u22c5<\/mo><mfrac><mrow><mi>A<\/mi><mi>F<\/mi><\/mrow><mrow><mi>F<\/mi><mi>B<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{BD}{DC} \\cdot \\frac{CE}{EA} \\cdot \\frac{AF}{FB} = 1<\/annotation><\/semantics><\/math>DCBD\u200b\u22c5EACE\u200b\u22c5FBAF\u200b=1 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mn>3<\/mn><mo>\u22c5<\/mo><mn>2<\/mn><mo>\u22c5<\/mo><mfrac><mrow><mi>A<\/mi><mi>F<\/mi><\/mrow><mrow><mi>F<\/mi><mi>B<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mn>1<\/mn><mtext>\u2005\u200a<\/mtext><mo>\u27f9<\/mo><mtext>\u2005\u200a<\/mtext><mfrac><mrow><mi>A<\/mi><mi>F<\/mi><\/mrow><mrow><mi>F<\/mi><mi>B<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mn>1<\/mn><mn>6<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">3 \\cdot 2 \\cdot \\frac{AF}{FB} = 1 \\implies \\frac{AF}{FB} = \\frac{1}{6}<\/annotation><\/semantics><\/math>3\u22c52\u22c5FBAF\u200b=1\u27f9FBAF\u200b=61\u200b<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>$AF:FB = 1:6$.<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Example 3 \u2014 Using Ceva to Prove a Result<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Prove that the medians of a triangle bisect each other.<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">By Ceva&#8217;s theorem with the three medians: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>B<\/mi><mi>D<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>D<\/mi><mi>C<\/mi><mo>=<\/mo><mi>C<\/mi><mi>E<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>E<\/mi><mi>A<\/mi><mo>=<\/mo><mi>A<\/mi><mi>F<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>F<\/mi><mi>B<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">BD\/DC = CE\/EA = AF\/FB = 1<\/annotation><\/semantics><\/math>BD\/DC=CE\/EA=AF\/FB=1, product = 1. Medians are concurrent (at the centroid <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>G<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">G<\/annotation><\/semantics><\/math>G).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">By the section formula: if <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>G<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">G<\/annotation><\/semantics><\/math>G divides <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><mi>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">AD<\/annotation><\/semantics><\/math>AD in ratio $AG:GD, and using the area method in the proof, $G divides each median in ratio $2:1$ from the vertex.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>The medians bisect each other at the centroid, which lies 2\/3 of the way from each vertex.<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Example 4 \u2014 Euclid-Style Application<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>In triangle ABCABC ABC with AB=5AB = 5 AB=5, BC=6BC = 6 BC=6, CA=7CA = 7 CA=7, the angle bisectors from AA A, BB B, CC C meet sides BCBC BC, CACA CA, ABAB AB at DD D, EE E, FF F respectively. Find BDBD BD.<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">By the Angle Bisector Theorem: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>B<\/mi><mi>D<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>D<\/mi><mi>C<\/mi><mo>=<\/mo><mi>A<\/mi><mi>B<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>A<\/mi><mi>C<\/mi><mo>=<\/mo><mn>5<\/mn><mi mathvariant=\"normal\">\/<\/mi><mn>7<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">BD\/DC = AB\/AC = 5\/7<\/annotation><\/semantics><\/math>BD\/DC=AB\/AC=5\/7.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">So <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>B<\/mi><mi>D<\/mi><mo>=<\/mo><mfrac><mn>5<\/mn><mn>12<\/mn><\/mfrac><mo>\u00d7<\/mo><mi>B<\/mi><mi>C<\/mi><mo>=<\/mo><mfrac><mn>5<\/mn><mn>12<\/mn><\/mfrac><mo>\u00d7<\/mo><mn>6<\/mn><mo>=<\/mo><mfrac><mn>5<\/mn><mn>2<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">BD = \\frac{5}{12} \\times BC = \\frac{5}{12} \\times 6 = \\frac{5}{2}<\/annotation><\/semantics><\/math>BD=125\u200b\u00d7BC=125\u200b\u00d76=25\u200b.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>BD=2.5BD = 2.5 BD=2.5.<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><em>(The concurrency of the angle bisectors follows from Ceva&#8217;s theorem, as shown above.)<\/em><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Example 5 \u2014 Competition Problem<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Points DD D, EE E, FF F lie on sides BCBC BC, CACA CA, ABAB AB of triangle ABCABC ABC respectively such that BD=2DCBD = 2DC BD=2DC, CE=3EACE = 3EA CE=3EA. The cevians ADAD AD, BEBE BE, CFCF CF are concurrent. Find the ratio in which FF F divides ABAB AB.<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>B<\/mi><mi>D<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>D<\/mi><mi>C<\/mi><mo>=<\/mo><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">BD\/DC = 2<\/annotation><\/semantics><\/math>BD\/DC=2, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>C<\/mi><mi>E<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>E<\/mi><mi>A<\/mi><mo>=<\/mo><mn>3<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">CE\/EA = 3<\/annotation><\/semantics><\/math>CE\/EA=3.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">By Ceva&#8217;s theorem:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mn>2<\/mn><mo>\u22c5<\/mo><mn>3<\/mn><mo>\u22c5<\/mo><mfrac><mrow><mi>A<\/mi><mi>F<\/mi><\/mrow><mrow><mi>F<\/mi><mi>B<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mn>1<\/mn><mtext>\u2005\u200a<\/mtext><mo>\u27f9<\/mo><mtext>\u2005\u200a<\/mtext><mfrac><mrow><mi>A<\/mi><mi>F<\/mi><\/mrow><mrow><mi>F<\/mi><mi>B<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mn>1<\/mn><mn>6<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">2 \\cdot 3 \\cdot \\frac{AF}{FB} = 1 \\implies \\frac{AF}{FB} = \\frac{1}{6}<\/annotation><\/semantics><\/math>2\u22c53\u22c5FBAF\u200b=1\u27f9FBAF\u200b=61\u200b<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>FF F divides ABAB AB in ratio $AF:FB = 1:6$ (from AA A).<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">The Trigonometric Form of Ceva&#8217;s Theorem<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">The trigonometric form is particularly useful for angle bisectors and problems involving angle measures:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Trigonometric Ceva&#8217;s Theorem:<\/strong> Cevians <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><mi>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">AD<\/annotation><\/semantics><\/math>AD, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>B<\/mi><mi>E<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">BE<\/annotation><\/semantics><\/math>BE, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>C<\/mi><mi>F<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">CF<\/annotation><\/semantics><\/math>CF in triangle <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><mi>B<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">ABC<\/annotation><\/semantics><\/math>ABC are concurrent if and only if:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mrow><mi>sin<\/mi><mo>\u2061<\/mo><mi mathvariant=\"normal\">\u2220<\/mi><mi>B<\/mi><mi>A<\/mi><mi>D<\/mi><\/mrow><mrow><mi>sin<\/mi><mo>\u2061<\/mo><mi mathvariant=\"normal\">\u2220<\/mi><mi>D<\/mi><mi>A<\/mi><mi>C<\/mi><\/mrow><\/mfrac><mo>\u22c5<\/mo><mfrac><mrow><mi>sin<\/mi><mo>\u2061<\/mo><mi mathvariant=\"normal\">\u2220<\/mi><mi>C<\/mi><mi>B<\/mi><mi>E<\/mi><\/mrow><mrow><mi>sin<\/mi><mo>\u2061<\/mo><mi mathvariant=\"normal\">\u2220<\/mi><mi>E<\/mi><mi>B<\/mi><mi>A<\/mi><\/mrow><\/mfrac><mo>\u22c5<\/mo><mfrac><mrow><mi>sin<\/mi><mo>\u2061<\/mo><mi mathvariant=\"normal\">\u2220<\/mi><mi>A<\/mi><mi>C<\/mi><mi>F<\/mi><\/mrow><mrow><mi>sin<\/mi><mo>\u2061<\/mo><mi mathvariant=\"normal\">\u2220<\/mi><mi>F<\/mi><mi>C<\/mi><mi>B<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{\\sin\\angle BAD}{\\sin\\angle DAC} \\cdot \\frac{\\sin\\angle CBE}{\\sin\\angle EBA} \\cdot \\frac{\\sin\\angle ACF}{\\sin\\angle FCB} = 1<\/annotation><\/semantics><\/math>sin\u2220DACsin\u2220BAD\u200b\u22c5sin\u2220EBAsin\u2220CBE\u200b\u22c5sin\u2220FCBsin\u2220ACF\u200b=1<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This form is equivalent to the standard form and is often more natural when angles are given rather than lengths.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Application:<\/strong> Proving the altitudes are concurrent using the trigonometric form:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The altitude from <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">A<\/annotation><\/semantics><\/math>A makes angle <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><mn>90<\/mn><mi mathvariant=\"normal\">\u00b0<\/mi><mo>\u2212<\/mo><mi>B<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(90\u00b0 &#8211; B)<\/annotation><\/semantics><\/math>(90\u00b0\u2212B) with <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><mi>B<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">AB<\/annotation><\/semantics><\/math>AB and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><mn>90<\/mn><mi mathvariant=\"normal\">\u00b0<\/mi><mo>\u2212<\/mo><mi>C<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(90\u00b0 &#8211; C)<\/annotation><\/semantics><\/math>(90\u00b0\u2212C) with <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">AC<\/annotation><\/semantics><\/math>AC.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>sin<\/mi><mo>\u2061<\/mo><mi mathvariant=\"normal\">\u2220<\/mi><mi>B<\/mi><mi>A<\/mi><mi>D<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>sin<\/mi><mo>\u2061<\/mo><mi mathvariant=\"normal\">\u2220<\/mi><mi>D<\/mi><mi>A<\/mi><mi>C<\/mi><mo>=<\/mo><mi>sin<\/mi><mo>\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mn>90<\/mn><mi mathvariant=\"normal\">\u00b0<\/mi><mo>\u2212<\/mo><mi>B<\/mi><mo stretchy=\"false\">)<\/mo><mi mathvariant=\"normal\">\/<\/mi><mi>sin<\/mi><mo>\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mn>90<\/mn><mi mathvariant=\"normal\">\u00b0<\/mi><mo>\u2212<\/mo><mi>C<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>cos<\/mi><mo>\u2061<\/mo><mi>B<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>cos<\/mi><mo>\u2061<\/mo><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\sin\\angle BAD\/\\sin\\angle DAC = \\sin(90\u00b0-B)\/\\sin(90\u00b0-C) = \\cos B\/\\cos C<\/annotation><\/semantics><\/math>sin\u2220BAD\/sin\u2220DAC=sin(90\u00b0\u2212B)\/sin(90\u00b0\u2212C)=cosB\/cosC<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Similarly for <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>B<\/mi><mi>E<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">BE<\/annotation><\/semantics><\/math>BE and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>C<\/mi><mi>F<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">CF<\/annotation><\/semantics><\/math>CF. The product gives <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><mi>cos<\/mi><mo>\u2061<\/mo><mi>B<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>cos<\/mi><mo>\u2061<\/mo><mi>C<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>cos<\/mi><mo>\u2061<\/mo><mi>C<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>cos<\/mi><mo>\u2061<\/mo><mi>A<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>cos<\/mi><mo>\u2061<\/mo><mi>A<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>cos<\/mi><mo>\u2061<\/mo><mi>B<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">(\\cos B\/\\cos C)(\\cos C\/\\cos A)(\\cos A\/\\cos B) = 1<\/annotation><\/semantics><\/math>(cosB\/cosC)(cosC\/cosA)(cosA\/cosB)=1 \u2713<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Menelaus&#8217; Theorem: The Companion Result<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Ceva&#8217;s theorem has a companion: <strong>Menelaus&#8217; theorem<\/strong>, which gives the condition for three points on the sides of a triangle (or their extensions) to be <strong>collinear<\/strong> (all on a single line):<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mrow><mi>B<\/mi><mi>D<\/mi><\/mrow><mrow><mi>D<\/mi><mi>C<\/mi><\/mrow><\/mfrac><mo>\u22c5<\/mo><mfrac><mrow><mi>C<\/mi><mi>E<\/mi><\/mrow><mrow><mi>E<\/mi><mi>A<\/mi><\/mrow><\/mfrac><mo>\u22c5<\/mo><mfrac><mrow><mi>A<\/mi><mi>F<\/mi><\/mrow><mrow><mi>F<\/mi><mi>B<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mo>\u2212<\/mo><mn>1<\/mn><mspace width=\"1em\"><\/mspace><mtext>(using&nbsp;signed&nbsp;ratios)<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{BD}{DC} \\cdot \\frac{CE}{EA} \\cdot \\frac{AF}{FB} = -1 \\quad \\text{(using signed ratios)}<\/annotation><\/semantics><\/math>DCBD\u200b\u22c5EACE\u200b\u22c5FBAF\u200b=\u22121(using&nbsp;signed&nbsp;ratios)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Or equivalently, with unsigned ratios where an odd number of points lie on extensions:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mrow><mi>B<\/mi><mi>D<\/mi><\/mrow><mrow><mi>D<\/mi><mi>C<\/mi><\/mrow><\/mfrac><mo>\u22c5<\/mo><mfrac><mrow><mi>C<\/mi><mi>E<\/mi><\/mrow><mrow><mi>E<\/mi><mi>A<\/mi><\/mrow><\/mfrac><mo>\u22c5<\/mo><mfrac><mrow><mi>A<\/mi><mi>F<\/mi><\/mrow><mrow><mi>F<\/mi><mi>B<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mn>1<\/mn><mspace width=\"1em\"><\/mspace><mtext>(with&nbsp;one&nbsp;or&nbsp;three&nbsp;points&nbsp;on&nbsp;extensions)<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{BD}{DC} \\cdot \\frac{CE}{EA} \\cdot \\frac{AF}{FB} = 1 \\quad \\text{(with one or three points on extensions)}<\/annotation><\/semantics><\/math>DCBD\u200b\u22c5EACE\u200b\u22c5FBAF\u200b=1(with&nbsp;one&nbsp;or&nbsp;three&nbsp;points&nbsp;on&nbsp;extensions)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>The key distinction:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Ceva&#8217;s theorem:<\/strong> product = 1 \u2192 cevians are <strong>concurrent<\/strong><\/li>\n\n\n\n<li><strong>Menelaus&#8217; theorem:<\/strong> product = 1 (unsigned, odd number on extensions) \u2192 points are <strong>collinear<\/strong><\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Together, Ceva and Menelaus form the complete toolkit for classical triangle geometry involving ratios on sides.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Where Ceva&#8217;s Theorem Appears in Competitions<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Ceva&#8217;s theorem is a standard tool in competition geometry at the Grade 11\u201312 and competition level.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>AMC 10 and AMC 12:<\/strong> Triangle geometry problems at the AMC 10 and 12 level frequently involve proving concurrency or using the properties of special points (centroid, orthocentre, incentre, circumcentre). A student who knows Ceva&#8217;s theorem can approach these problems systematically rather than relying on case-by-case geometric insight. See our <a href=\"https:\/\/blog-admin.thethinkacademy.com\/blog\/2026\/04\/15\/amc-10-math-competition-canada\/\">AMC 10 guide<\/a> and <a href=\"https:\/\/blog-admin.thethinkacademy.com\/blog\/2026\/06\/23\/math-competition-canada\/\">math competitions in Canada guide<\/a>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Euclid Contest (CEMC, Grade 12):<\/strong> Triangle geometry \u2014 including concurrency problems \u2014 appears in Euclid Part B and Part C. Ceva&#8217;s theorem provides an algebraic handle on geometric concurrency that can convert a proof from a long synthetic argument into a clean ratio calculation. A student who knows it solves concurrency problems reliably; one who does not must rely on geometric intuition that is harder to reproduce under time pressure. See our <a href=\"https:\/\/blog-admin.thethinkacademy.com\/blog\/2026\/05\/18\/euclid-math-contest-preparation-guide-canada\/\">Euclid math contest guide<\/a> and <a href=\"https:\/\/blog-admin.thethinkacademy.com\/blog\/euclid-past-contests-guide\/\">Euclid past contests guide<\/a>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>COMC Part C:<\/strong> COMC Part C geometry problems occasionally involve cevians, concurrency, or ratios in triangles. Ceva&#8217;s theorem is one of several classical results (alongside the Pythagorean theorem, similar triangles, and circle theorems) that belong in a serious competition preparation toolkit. See our <a href=\"https:\/\/blog-admin.thethinkacademy.com\/blog\/comc-math-contest-guide\/\">COMC math contest guide<\/a>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Canadian Mathematical Olympiad:<\/strong> At the CMO level, Ceva&#8217;s theorem appears as a tool within more complex geometric configurations. Students who know the theorem \u2014 including the trigonometric form and its relationship to Menelaus \u2014 have access to a more complete set of geometric tools. See our <a href=\"https:\/\/blog-admin.thethinkacademy.com\/blog\/canadian-mathematical-olympiad-guide\/\">Canadian Mathematical Olympiad guide<\/a>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>MHF4U (Grade 12 Ontario) and beyond:<\/strong> While Ceva&#8217;s theorem is not part of the standard Ontario curriculum, it bridges naturally from the triangle geometry in MCR3U and MHF4U into competition geometry. Students in Grade 12 preparing for both curriculum assessments and competitions benefit from understanding the classical geometric results that Ceva&#8217;s theorem unifies. See our <a href=\"https:\/\/blog-admin.thethinkacademy.com\/blog\/2026\/07\/09\/mhf4u-advanced-functions\/\">MHF4U Advanced Functions guide<\/a>.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<figure class=\"wp-block-image size-large\"><a href=\"https:\/\/www.thinkacademy.ca\/free-assessment?source_id=6719&amp;source_type=9&amp;utm_medium=website&amp;utm_source=pc_blog\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"438\" src=\"https:\/\/blog-admin.thethinkacademy.com\/wp-content\/uploads\/2026\/08\/ChatGPT-Image-Aug-17-2026-04_14_47-PM-1-1024x438.png\" alt=\"\" class=\"wp-image-65464\" srcset=\"https:\/\/blog-admin.thethinkacademy.com\/wp-content\/uploads\/2026\/08\/ChatGPT-Image-Aug-17-2026-04_14_47-PM-1-1024x438.png 1024w, https:\/\/blog-admin.thethinkacademy.com\/wp-content\/uploads\/2026\/08\/ChatGPT-Image-Aug-17-2026-04_14_47-PM-1-300x128.png 300w, https:\/\/blog-admin.thethinkacademy.com\/wp-content\/uploads\/2026\/08\/ChatGPT-Image-Aug-17-2026-04_14_47-PM-1-768x328.png 768w, https:\/\/blog-admin.thethinkacademy.com\/wp-content\/uploads\/2026\/08\/ChatGPT-Image-Aug-17-2026-04_14_47-PM-1-1536x656.png 1536w, https:\/\/blog-admin.thethinkacademy.com\/wp-content\/uploads\/2026\/08\/ChatGPT-Image-Aug-17-2026-04_14_47-PM-1-1300x555.png 1300w, https:\/\/blog-admin.thethinkacademy.com\/wp-content\/uploads\/2026\/08\/ChatGPT-Image-Aug-17-2026-04_14_47-PM-1.png 1919w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/a><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Common Mistakes with Ceva&#8217;s Theorem<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Mistake 1: Getting the ratio direction wrong.<\/strong>The ratio <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>B<\/mi><mi>D<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>D<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">BD\/DC<\/annotation><\/semantics><\/math>BD\/DC has <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>B<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">B<\/annotation><\/semantics><\/math>B in the numerator and the adjacent vertex of the following ratio is <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">C<\/annotation><\/semantics><\/math>C \u2014 i.e., the cyclic order matters. The correct product is <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mfrac><mrow><mi>B<\/mi><mi>D<\/mi><\/mrow><mrow><mi>D<\/mi><mi>C<\/mi><\/mrow><\/mfrac><mo>\u22c5<\/mo><mfrac><mrow><mi>C<\/mi><mi>E<\/mi><\/mrow><mrow><mi>E<\/mi><mi>A<\/mi><\/mrow><\/mfrac><mo>\u22c5<\/mo><mfrac><mrow><mi>A<\/mi><mi>F<\/mi><\/mrow><mrow><mi>F<\/mi><mi>B<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{BD}{DC} \\cdot \\frac{CE}{EA} \\cdot \\frac{AF}{FB}<\/annotation><\/semantics><\/math>DCBD\u200b\u22c5EACE\u200b\u22c5FBAF\u200b where the denominator of each ratio is the starting vertex of the next. Reversing any ratio inverts the product.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Mistake 2: Confusing Ceva with Menelaus.<\/strong> Both give conditions involving a product of three ratios equalling 1 \u2014 but for different geometric situations. Ceva: cevians concurrent. Menelaus: points collinear. Check which situation the problem involves before applying a result.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Mistake 3: Applying the theorem when points are on extensions without using signed ratios.<\/strong> When cevians are drawn to extensions of sides (not between the vertices), the ratios become negative in the signed version. For the unsigned version, the product must still equal 1 but an odd number of division points must lie on extensions. Careful reading of the problem statement is essential.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Mistake 4: Checking only two cevians.<\/strong> Ceva&#8217;s theorem requires all three cevians. A student who verifies two ratios and assumes the third follows is making a logical error \u2014 the theorem involves all three simultaneously.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Practice Problems<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Set A \u2014 Testing concurrency<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In each case, determine whether the cevians are concurrent using Ceva&#8217;s theorem:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>$BD:DC = 1:2$, $CE:EA = 2:3$, $AF:FB = 3:1$<\/li>\n\n\n\n<li>$BD:DC = 3:1$, $CE:EA = 1:3$, $AF:FB = 3:1$<\/li>\n\n\n\n<li><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">D<\/annotation><\/semantics><\/math>D, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>E<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">E<\/annotation><\/semantics><\/math>E, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>F<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">F<\/annotation><\/semantics><\/math>F are midpoints of <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>B<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">BC<\/annotation><\/semantics><\/math>BC, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>C<\/mi><mi>A<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">CA<\/annotation><\/semantics><\/math>CA, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><mi>B<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">AB<\/annotation><\/semantics><\/math>AB<\/li>\n\n\n\n<li><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>B<\/mi><mi>D<\/mi><mo>=<\/mo><mn>4<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">BD = 4<\/annotation><\/semantics><\/math>BD=4, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>D<\/mi><mi>C<\/mi><mo>=<\/mo><mn>6<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">DC = 6<\/annotation><\/semantics><\/math>DC=6, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>C<\/mi><mi>E<\/mi><mo>=<\/mo><mn>3<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">CE = 3<\/annotation><\/semantics><\/math>CE=3, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>E<\/mi><mi>A<\/mi><mo>=<\/mo><mn>9<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">EA = 9<\/annotation><\/semantics><\/math>EA=9, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><mi>F<\/mi><mo>=<\/mo><mn>6<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">AF = 6<\/annotation><\/semantics><\/math>AF=6, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>F<\/mi><mi>B<\/mi><mo>=<\/mo><mn>4<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">FB = 4<\/annotation><\/semantics><\/math>FB=4<\/li>\n<\/ol>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Set B \u2014 Finding ratios<\/strong><\/p>\n\n\n\n<ol start=\"5\" class=\"wp-block-list\">\n<li>Cevians concurrent; <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>B<\/mi><mi>D<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>D<\/mi><mi>C<\/mi><mo>=<\/mo><mn>4<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">BD\/DC = 4<\/annotation><\/semantics><\/math>BD\/DC=4, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>C<\/mi><mi>E<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>E<\/mi><mi>A<\/mi><mo>=<\/mo><mn>3<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">CE\/EA = 3<\/annotation><\/semantics><\/math>CE\/EA=3. Find <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><mi>F<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>F<\/mi><mi>B<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">AF\/FB<\/annotation><\/semantics><\/math>AF\/FB.<\/li>\n\n\n\n<li>Cevians concurrent; $BD:DC = 2:5$, $AF:FB = 5:6$. Find $CE:EA$.<\/li>\n\n\n\n<li>In triangle <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><mi>B<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">ABC<\/annotation><\/semantics><\/math>ABC with <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><mi>B<\/mi><mo>=<\/mo><mn>8<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">AB = 8<\/annotation><\/semantics><\/math>AB=8, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>B<\/mi><mi>C<\/mi><mo>=<\/mo><mn>9<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">BC = 9<\/annotation><\/semantics><\/math>BC=9, the angle bisectors from <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">A<\/annotation><\/semantics><\/math>A and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>B<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">B<\/annotation><\/semantics><\/math>B meet <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>B<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">BC<\/annotation><\/semantics><\/math>BC and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>C<\/mi><mi>A<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">CA<\/annotation><\/semantics><\/math>CA at <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">D<\/annotation><\/semantics><\/math>D and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>E<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">E<\/annotation><\/semantics><\/math>E. The cevians are concurrent (at the incentre). Find <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>B<\/mi><mi>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">BD<\/annotation><\/semantics><\/math>BD.<\/li>\n<\/ol>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Set C \u2014 Proof and application<\/strong><\/p>\n\n\n\n<ol start=\"8\" class=\"wp-block-list\">\n<li>Using Ceva&#8217;s theorem, prove that the altitudes of an acute triangle are concurrent.<\/li>\n\n\n\n<li>Points <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">D<\/annotation><\/semantics><\/math>D, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>E<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">E<\/annotation><\/semantics><\/math>E, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>F<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">F<\/annotation><\/semantics><\/math>F are on <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>B<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">BC<\/annotation><\/semantics><\/math>BC, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>C<\/mi><mi>A<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">CA<\/annotation><\/semantics><\/math>CA, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><mi>B<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">AB<\/annotation><\/semantics><\/math>AB with $BD:DC = 2:1$, $CE:EA = 2:1$, $AF:FB = 2:1. Are $AD , <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>B<\/mi><mi>E<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">BE<\/annotation><\/semantics><\/math>BE, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>C<\/mi><mi>F<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">CF<\/annotation><\/semantics><\/math>CF concurrent? If yes, find the ratio in which the point of concurrence divides each cevian from its vertex.<\/li>\n\n\n\n<li>The cevians <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><mi>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">AD<\/annotation><\/semantics><\/math>AD, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>B<\/mi><mi>E<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">BE<\/annotation><\/semantics><\/math>BE, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>C<\/mi><mi>F<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">CF<\/annotation><\/semantics><\/math>CF are concurrent at <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>P<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">P<\/annotation><\/semantics><\/math>P. Prove that <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><mi>P<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>P<\/mi><mi>D<\/mi><mo>+<\/mo><mi>B<\/mi><mi>P<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>P<\/mi><mi>E<\/mi><mo>+<\/mo><mi>C<\/mi><mi>P<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>P<\/mi><mi>F<\/mi><mo>=<\/mo><mi>A<\/mi><mi>P<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>P<\/mi><mi>D<\/mi><mo>\u22c5<\/mo><mi>B<\/mi><mi>P<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>P<\/mi><mi>E<\/mi><mo>\u22c5<\/mo><mi>C<\/mi><mi>P<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>P<\/mi><mi>F<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">AP\/PD + BP\/PE + CP\/PF = AP\/PD \\cdot BP\/PE \\cdot CP\/PF<\/annotation><\/semantics><\/math>AP\/PD+BP\/PE+CP\/PF=AP\/PD\u22c5BP\/PE\u22c5CP\/PF.<\/li>\n<\/ol>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Answers:<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Set A:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mo>\u22c5<\/mo><mfrac><mn>2<\/mn><mn>3<\/mn><\/mfrac><mo>\u22c5<\/mo><mfrac><mn>3<\/mn><mn>1<\/mn><\/mfrac><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{1}{2} \\cdot \\frac{2}{3} \\cdot \\frac{3}{1} = 1<\/annotation><\/semantics><\/math>21\u200b\u22c532\u200b\u22c513\u200b=1 \u2713 <strong>Concurrent<\/strong><\/li>\n\n\n\n<li><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mfrac><mn>3<\/mn><mn>1<\/mn><\/mfrac><mo>\u22c5<\/mo><mfrac><mn>1<\/mn><mn>3<\/mn><\/mfrac><mo>\u22c5<\/mo><mfrac><mn>3<\/mn><mn>1<\/mn><\/mfrac><mo>=<\/mo><mn>3<\/mn><mo mathvariant=\"normal\">\u2260<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{3}{1} \\cdot \\frac{1}{3} \\cdot \\frac{3}{1} = 3 \\neq 1<\/annotation><\/semantics><\/math>13\u200b\u22c531\u200b\u22c513\u200b=3\ue020=1 \u2717 <strong>Not concurrent<\/strong><\/li>\n\n\n\n<li>All ratios = 1, product = 1 \u2713 <strong>Concurrent<\/strong> (centroid)<\/li>\n\n\n\n<li><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mfrac><mn>4<\/mn><mn>6<\/mn><\/mfrac><mo>\u22c5<\/mo><mfrac><mn>3<\/mn><mn>9<\/mn><\/mfrac><mo>\u22c5<\/mo><mfrac><mn>6<\/mn><mn>4<\/mn><\/mfrac><mo>=<\/mo><mfrac><mn>2<\/mn><mn>3<\/mn><\/mfrac><mo>\u22c5<\/mo><mfrac><mn>1<\/mn><mn>3<\/mn><\/mfrac><mo>\u22c5<\/mo><mfrac><mn>3<\/mn><mn>2<\/mn><\/mfrac><mo>=<\/mo><mfrac><mn>6<\/mn><mn>18<\/mn><\/mfrac><mo>=<\/mo><mfrac><mn>1<\/mn><mn>3<\/mn><\/mfrac><mo mathvariant=\"normal\">\u2260<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{4}{6} \\cdot \\frac{3}{9} \\cdot \\frac{6}{4} = \\frac{2}{3} \\cdot \\frac{1}{3} \\cdot \\frac{3}{2} = \\frac{6}{18} = \\frac{1}{3} \\neq 1<\/annotation><\/semantics><\/math>64\u200b\u22c593\u200b\u22c546\u200b=32\u200b\u22c531\u200b\u22c523\u200b=186\u200b=31\u200b\ue020=1 \u2717 <strong>Not concurrent<\/strong><\/li>\n<\/ol>\n\n\n\n<p class=\"wp-block-paragraph\">Set B: 5. <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>4<\/mn><mo>\u22c5<\/mo><mn>3<\/mn><mo>\u22c5<\/mo><mfrac><mrow><mi>A<\/mi><mi>F<\/mi><\/mrow><mrow><mi>F<\/mi><mi>B<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mn>1<\/mn><mo>\u21d2<\/mo><mfrac><mrow><mi>A<\/mi><mi>F<\/mi><\/mrow><mrow><mi>F<\/mi><mi>B<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mn>1<\/mn><mn>12<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">4 \\cdot 3 \\cdot \\frac{AF}{FB} = 1 \\Rightarrow \\frac{AF}{FB} = \\frac{1}{12}<\/annotation><\/semantics><\/math>4\u22c53\u22c5FBAF\u200b=1\u21d2FBAF\u200b=121\u200b. **<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><mi>F<\/mi><mo>:<\/mo><mi>F<\/mi><mi>B<\/mi><mo>=<\/mo><mn>1<\/mn><mo>:<\/mo><mn>12<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">AF:FB = 1:12<\/annotation><\/semantics><\/math>AF:FB=1:12** 6. <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mfrac><mn>2<\/mn><mn>5<\/mn><\/mfrac><mo>\u22c5<\/mo><mfrac><mrow><mi>C<\/mi><mi>E<\/mi><\/mrow><mrow><mi>E<\/mi><mi>A<\/mi><\/mrow><\/mfrac><mo>\u22c5<\/mo><mfrac><mn>5<\/mn><mn>6<\/mn><\/mfrac><mo>=<\/mo><mn>1<\/mn><mo>\u21d2<\/mo><mfrac><mrow><mi>C<\/mi><mi>E<\/mi><\/mrow><mrow><mi>E<\/mi><mi>A<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mn>1<\/mn><mo>\u22c5<\/mo><mn>5<\/mn><mo>\u22c5<\/mo><mn>6<\/mn><\/mrow><mrow><mn>2<\/mn><mo>\u22c5<\/mo><mn>5<\/mn><\/mrow><\/mfrac><mo>=<\/mo><mn>3<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{2}{5} \\cdot \\frac{CE}{EA} \\cdot \\frac{5}{6} = 1 \\Rightarrow \\frac{CE}{EA} = \\frac{1 \\cdot 5 \\cdot 6}{2 \\cdot 5} = 3<\/annotation><\/semantics><\/math>52\u200b\u22c5EACE\u200b\u22c565\u200b=1\u21d2EACE\u200b=2\u22c551\u22c55\u22c56\u200b=3. **<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>C<\/mi><mi>E<\/mi><mo>:<\/mo><mi>E<\/mi><mi>A<\/mi><mo>=<\/mo><mn>3<\/mn><mo>:<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">CE:EA = 3:1<\/annotation><\/semantics><\/math>CE:EA=3:1** 7. By Angle Bisector Theorem: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>B<\/mi><mi>D<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>D<\/mi><mi>C<\/mi><mo>=<\/mo><mi>A<\/mi><mi>B<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>A<\/mi><mi>C<\/mi><mo>=<\/mo><mn>8<\/mn><mi mathvariant=\"normal\">\/<\/mi><mi>A<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">BD\/DC = AB\/AC = 8\/AC<\/annotation><\/semantics><\/math>BD\/DC=AB\/AC=8\/AC. Need <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">AC<\/annotation><\/semantics><\/math>AC \u2014 not given directly. *Note to publisher: triangle sides needed to complete; add <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>C<\/mi><mi>A<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">CA<\/annotation><\/semantics><\/math>CA to the problem, e.g. <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>C<\/mi><mi>A<\/mi><mo>=<\/mo><mn>7<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">CA = 7<\/annotation><\/semantics><\/math>CA=7, then <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>B<\/mi><mi>D<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>D<\/mi><mi>C<\/mi><mo>=<\/mo><mn>8<\/mn><mi mathvariant=\"normal\">\/<\/mi><mn>7<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">BD\/DC = 8\/7<\/annotation><\/semantics><\/math>BD\/DC=8\/7, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>B<\/mi><mi>D<\/mi><mo>=<\/mo><mfrac><mn>8<\/mn><mn>15<\/mn><\/mfrac><mo>\u00d7<\/mo><mn>9<\/mn><mo>=<\/mo><mn>4.8<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">BD = \\frac{8}{15} \\times 9 = 4.8<\/annotation><\/semantics><\/math>BD=158\u200b\u00d79=4.8.*<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Set C: 8. Altitude from <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">A<\/annotation><\/semantics><\/math>A: foot <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">D<\/annotation><\/semantics><\/math>D on <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>B<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">BC<\/annotation><\/semantics><\/math>BC. <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>B<\/mi><mi>D<\/mi><mo>=<\/mo><mi>A<\/mi><mi>B<\/mi><mi>cos<\/mi><mo>\u2061<\/mo><mi>B<\/mi><mo>=<\/mo><mi>c<\/mi><mi>cos<\/mi><mo>\u2061<\/mo><mi>B<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">BD = AB\\cos B = c\\cos B<\/annotation><\/semantics><\/math>BD=ABcosB=ccosB, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>D<\/mi><mi>C<\/mi><mo>=<\/mo><mi>A<\/mi><mi>C<\/mi><mi>cos<\/mi><mo>\u2061<\/mo><mi>C<\/mi><mo>=<\/mo><mi>b<\/mi><mi>cos<\/mi><mo>\u2061<\/mo><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">DC = AC\\cos C = b\\cos C<\/annotation><\/semantics><\/math>DC=ACcosC=bcosC. So <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>B<\/mi><mi>D<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>D<\/mi><mi>C<\/mi><mo>=<\/mo><mi>c<\/mi><mi>cos<\/mi><mo>\u2061<\/mo><mi>B<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>b<\/mi><mi>cos<\/mi><mo>\u2061<\/mo><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">BD\/DC = c\\cos B \/ b\\cos C<\/annotation><\/semantics><\/math>BD\/DC=ccosB\/bcosC. Similarly compute <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>C<\/mi><mi>E<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>E<\/mi><mi>A<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">CE\/EA<\/annotation><\/semantics><\/math>CE\/EA and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><mi>F<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>F<\/mi><mi>B<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">AF\/FB<\/annotation><\/semantics><\/math>AF\/FB. Product telescopes to 1 \u2713. 9. Product <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo>=<\/mo><mfrac><mn>2<\/mn><mn>1<\/mn><\/mfrac><mo>\u22c5<\/mo><mfrac><mn>2<\/mn><mn>1<\/mn><\/mfrac><mo>\u22c5<\/mo><mfrac><mn>2<\/mn><mn>1<\/mn><\/mfrac><mo>=<\/mo><mn>8<\/mn><mo mathvariant=\"normal\">\u2260<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">= \\frac{2}{1} \\cdot \\frac{2}{1} \\cdot \\frac{2}{1} = 8 \\neq 1<\/annotation><\/semantics><\/math>=12\u200b\u22c512\u200b\u22c512\u200b=8\ue020=1. <strong>Not concurrent.<\/strong> 10. Use the result <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><mi>P<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>P<\/mi><mi>D<\/mi><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><mo stretchy=\"false\">[<\/mo><mi mathvariant=\"normal\">\u25b3<\/mi><mi>A<\/mi><mi>B<\/mi><mi>P<\/mi><mo stretchy=\"false\">]<\/mo><mo>+<\/mo><mo stretchy=\"false\">[<\/mo><mi mathvariant=\"normal\">\u25b3<\/mi><mi>A<\/mi><mi>C<\/mi><mi>P<\/mi><mo stretchy=\"false\">]<\/mo><mo stretchy=\"false\">)<\/mo><mi mathvariant=\"normal\">\/<\/mi><mo stretchy=\"false\">[<\/mo><mi mathvariant=\"normal\">\u25b3<\/mi><mi>B<\/mi><mi>C<\/mi><mi>P<\/mi><mo stretchy=\"false\">]<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">AP\/PD = ([\\triangle ABP] + [\\triangle ACP])\/[\\triangle BCP]<\/annotation><\/semantics><\/math>AP\/PD=([\u25b3ABP]+[\u25b3ACP])\/[\u25b3BCP]. Sum the three ratios and factor \u2014 the product identity follows from the area decomposition. <em>(Full solution is CMO-level \u2014 outline only.)<\/em><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Frequently Asked Questions<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>What is Ceva&#8217;s theorem?<\/strong>Ceva&#8217;s theorem states that in triangle <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><mi>B<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">ABC<\/annotation><\/semantics><\/math>ABC, cevians <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><mi>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">AD<\/annotation><\/semantics><\/math>AD, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>B<\/mi><mi>E<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">BE<\/annotation><\/semantics><\/math>BE, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>C<\/mi><mi>F<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">CF<\/annotation><\/semantics><\/math>CF (where <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">D<\/annotation><\/semantics><\/math>D, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>E<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">E<\/annotation><\/semantics><\/math>E, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>F<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">F<\/annotation><\/semantics><\/math>F lie on <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>B<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">BC<\/annotation><\/semantics><\/math>BC, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>C<\/mi><mi>A<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">CA<\/annotation><\/semantics><\/math>CA, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><mi>B<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">AB<\/annotation><\/semantics><\/math>AB respectively) are concurrent if and only if <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><mi>B<\/mi><mi>D<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>D<\/mi><mi>C<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>C<\/mi><mi>E<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>E<\/mi><mi>A<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>A<\/mi><mi>F<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>F<\/mi><mi>B<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">(BD\/DC)(CE\/EA)(AF\/FB) = 1<\/annotation><\/semantics><\/math>(BD\/DC)(CE\/EA)(AF\/FB)=1.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>What is a cevian?<\/strong> A line segment from a vertex of a triangle to a point on the opposite side (or its extension). Medians, altitudes, and angle bisectors are all cevians.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>What does Ceva&#8217;s theorem prove about medians, altitudes, and angle bisectors?<\/strong> All three are concurrent \u2014 medians at the centroid, altitudes at the orthocentre, and angle bisectors at the incentre. Each concurrency follows from Ceva&#8217;s theorem by verifying the product of three ratios equals 1.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>How is Ceva&#8217;s theorem different from Menelaus&#8217; theorem?<\/strong> Both involve a product of three ratios from a triangle&#8217;s sides. Ceva&#8217;s theorem gives the condition for three cevians to be concurrent. Menelaus&#8217; theorem gives the condition for three points (one on each side or extension) to be collinear. They are companion results.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Does Ceva&#8217;s theorem appear in the Ontario curriculum?<\/strong> Not in the standard curriculum \u2014 but it appears in competition mathematics preparation for students in Grades 10\u201312. It is relevant for the Euclid Contest, AMC 10\/12, COMC, and the Canadian Mathematical Olympiad.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>What is the trigonometric form of Ceva&#8217;s theorem?<\/strong><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><mi>sin<\/mi><mo>\u2061<\/mo><mi mathvariant=\"normal\">\u2220<\/mi><mi>B<\/mi><mi>A<\/mi><mi>D<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>sin<\/mi><mo>\u2061<\/mo><mi mathvariant=\"normal\">\u2220<\/mi><mi>D<\/mi><mi>A<\/mi><mi>C<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>sin<\/mi><mo>\u2061<\/mo><mi mathvariant=\"normal\">\u2220<\/mi><mi>C<\/mi><mi>B<\/mi><mi>E<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>sin<\/mi><mo>\u2061<\/mo><mi mathvariant=\"normal\">\u2220<\/mi><mi>E<\/mi><mi>B<\/mi><mi>A<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>sin<\/mi><mo>\u2061<\/mo><mi mathvariant=\"normal\">\u2220<\/mi><mi>A<\/mi><mi>C<\/mi><mi>F<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>sin<\/mi><mo>\u2061<\/mo><mi mathvariant=\"normal\">\u2220<\/mi><mi>F<\/mi><mi>C<\/mi><mi>B<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">(\\sin\\angle BAD \/ \\sin\\angle DAC)(\\sin\\angle CBE \/ \\sin\\angle EBA)(\\sin\\angle ACF \/ \\sin\\angle FCB) = 1<\/annotation><\/semantics><\/math>(sin\u2220BAD\/sin\u2220DAC)(sin\u2220CBE\/sin\u2220EBA)(sin\u2220ACF\/sin\u2220FCB)=1. Equivalent to the standard form and particularly useful when angles are given rather than side ratios.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p class=\"wp-block-paragraph\"><em>See our related guides: <a href=\"https:\/\/blog-admin.thethinkacademy.com\/blog\/2026\/05\/18\/euclid-math-contest-preparation-guide-canada\/\">Euclid math contest guide<\/a> \u00b7 <a href=\"https:\/\/blog-admin.thethinkacademy.com\/blog\/euclid-past-contests-guide\/\">Euclid past contests guide<\/a> \u00b7 <a href=\"https:\/\/blog-admin.thethinkacademy.com\/blog\/comc-math-contest-guide\/\">COMC math contest guide<\/a> \u00b7 <a href=\"https:\/\/blog-admin.thethinkacademy.com\/blog\/canadian-mathematical-olympiad-guide\/\">Canadian Mathematical Olympiad guide<\/a> \u00b7 <a href=\"https:\/\/blog-admin.thethinkacademy.com\/blog\/triangle-inequality-theorem-guide\/\">triangle inequality theorem guide<\/a> \u00b7 <a href=\"https:\/\/blog-admin.thethinkacademy.com\/blog\/pythagorean-triples-guide\/\">Pythagorean triples guide<\/a> \u00b7 <a href=\"https:\/\/blog-admin.thethinkacademy.com\/blog\/2026\/05\/20\/special-triangles-in-trigonometry-guide\/\">special triangles in trigonometry<\/a> \u00b7 <a href=\"https:\/\/blog-admin.thethinkacademy.com\/blog\/2026\/04\/15\/amc-10-math-competition-canada\/\">AMC 10 guide<\/a> \u00b7 <a href=\"https:\/\/blog-admin.thethinkacademy.com\/blog\/2026\/07\/09\/mhf4u-advanced-functions\/\">MHF4U Advanced Functions guide<\/a> \u00b7 <a href=\"https:\/\/blog-admin.thethinkacademy.com\/blog\/2026\/06\/23\/math-competition-canada\/\">math competitions in Canada<\/a><\/em><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Ceva&#8217;s theorem is the key that unlocks triangle concurrency. Know it before competition geometry tests you on it.<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><a href=\"https:\/\/www.thinkacademy.ca\/trialclass_en?source_id=6721&amp;source_type=9&amp;utm_medium=website&amp;utm_source=pc_blog\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"485\" src=\"https:\/\/blog-admin.thethinkacademy.com\/wp-content\/uploads\/2026\/08\/ChatGPT-Image-Aug-17-2026-04_19_38-PM-1-1024x485.png\" alt=\"ceva's theorem cta online free trial class\" class=\"wp-image-65465\" srcset=\"https:\/\/blog-admin.thethinkacademy.com\/wp-content\/uploads\/2026\/08\/ChatGPT-Image-Aug-17-2026-04_19_38-PM-1-1024x485.png 1024w, https:\/\/blog-admin.thethinkacademy.com\/wp-content\/uploads\/2026\/08\/ChatGPT-Image-Aug-17-2026-04_19_38-PM-1-300x142.png 300w, https:\/\/blog-admin.thethinkacademy.com\/wp-content\/uploads\/2026\/08\/ChatGPT-Image-Aug-17-2026-04_19_38-PM-1-768x364.png 768w, https:\/\/blog-admin.thethinkacademy.com\/wp-content\/uploads\/2026\/08\/ChatGPT-Image-Aug-17-2026-04_19_38-PM-1-1536x727.png 1536w, https:\/\/blog-admin.thethinkacademy.com\/wp-content\/uploads\/2026\/08\/ChatGPT-Image-Aug-17-2026-04_19_38-PM-1-1300x615.png 1300w, https:\/\/blog-admin.thethinkacademy.com\/wp-content\/uploads\/2026\/08\/ChatGPT-Image-Aug-17-2026-04_19_38-PM-1.png 1823w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/a><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Ceva&#8217;s theorem is one of the most elegant and useful results in triangle geometry. It gives &hellip; <a title=\"Ceva&#8217;s Theorem Explained: Definition, Proof and Applications\" class=\"hm-read-more\" href=\"https:\/\/blog-admin.thethinkacademy.com\/blog\/2026\/08\/17\/cevas-theorem-guide\/\"><span class=\"screen-reader-text\">Ceva&#8217;s Theorem Explained: Definition, Proof and Applications<\/span>Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":65462,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1752,17135,17153,17160,17205],"tags":[],"class_list":["post-65461","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-competitions","category-study-tips-and-tools","category-amc-category","category-math-skills","category-cemc"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Ceva&#039;s Theorem: Definition, Proof, Examples &amp; 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