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Like the Theorem of Menelaus, the Theorem of Ceva is often stated in terms of a triangle, but is in actuality a more general fact of plane geometry. The theorem and proof are taken from Dan Pedoe's book Geometry.


Let A, B, and C be vectors defining three points. Let D be any other point in the plane defined by those points. Form lines AB, BC, and AC that define the interior and exterior of triangle ABC. Form another set of lines AD, BD, and CD that intersect lines BC, AC, and AB at points L, M, and N. Then (BL/LC)(CM/MA)(AN/NB)= 1, where each two-letter pair is a signed length. If D is in the interior of the triangle than each of the lengths are positive and the theorem is slightly easier to visualize.


It's easy to show that L = xB + x'C, M = yC + y'A, and N = zA + z'B where x+x' = y+y' = z+z' = 1 and BL/LC = x'/x etc. D can be written uniquely as aA + bB + cC where a + b + c = 1 (see barycentric coordinates). This next part is exceedingly clever. Rewrite D as aA + (b+c)X, where X = (bB + cC)/(b+c). Then X lies on both AD and BC so it is L! Therefore x = b/(b+c) and x' = c/(b+c) and x/x' = b/c. Repeat the above three times for points B and C and the theorem is uncovered.

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