Ceva's Theorem

Ceva's theorem is a theorem about triangles in Euclidean plane geometry. Given a triangle ABC, let the lines AO, BO and CO be drawn from the vertices to a common point O to meet opposite sides at D, E and F respectively. Then

This equation uses signed lengths of segments, in other words the length AB is taken to be positive or negative according to whether A is to the left or right of B in some fixed orientation of the line. For example, AF/FB is defined as having positive value when F is between A and B and negative otherwise.

The converse is also true: If points D, E and F are chosen on BC, AC and AB respectively so that

then AD, BE and CF are concurrent. The converse is often included as part of the theorem.

The theorem is often attributed to Giovanni Ceva, who published it in his 1678 work De lineis rectis. But it was proven much earlier by Yusuf Al-Mu'taman ibn Hűd, an eleventh-century king of Zaragoza.

Associated with the figures are several terms derived from Ceva's name: cevian (the lines AD, BE, CF are the cevians of O), cevian triangle (the triangle DEF is the cevian triangle of O); cevian nest, anticevian triangle, Ceva conjugate. (Ceva is pronounced Chay'va; cevian is pronounced chev'ian.)

The theorem is very similar to Menelaus' theorem in that their equations differ only in sign.