Ceva's theorem

In Euclidean geometry, Ceva's theorem is a theorem about triangles. Given a triangle $△ ABC$ , let the lines $AO, BO, CO$ be drawn from the vertices to a common point $O$ (not on one of the sides of $△ ABC$ ), to meet opposite sides at $D, E, F$ respectively. (The segments $AD, BE, CF$ are known as cevians.) Then, using signed lengths of segments,

{\frac {\overline {AF}}{\overline {FB}}}\cdot {\frac {\overline {BD}}{\overline {DC}}}\cdot {\frac {\overline {CE}}{\overline {EA}}}=1.

In other words, the length $XY$ is taken to be positive or negative according to whether $X$ is to the left or right of $Y$ 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.

Ceva's theorem is a theorem of affine geometry, in the sense that it may be stated and proved without using the concepts of angles, areas, and lengths (except for the ratio of the lengths of two line segments that are collinear). It is therefore true for triangles in any affine plane over any field.

A slightly adapted converse is also true: If points $D, E, F$ are chosen on $BC, AC, AB$ respectively so that

{\frac {\overline {AF}}{\overline {FB}}}\cdot {\frac {\overline {BD}}{\overline {DC}}}\cdot {\frac {\overline {CE}}{\overline {EA}}}=1,

then $AD, BE, CF$ are concurrent, or all three parallel. 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.^[1]

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. By re-writing each in terms of cross-ratios, the two theorems may be seen as projective duals.^[2]

^ Holme, Audun (2010). Geometry: Our Cultural Heritage. Springer. p. 210. ISBN 978-3-642-14440-0.
^ Benitez, Julio (2007). "A Unified Proof of Ceva and Menelaus' Theorems Using Projective Geometry" (PDF). Journal for Geometry and Graphics. 11 (1): 39–44.

[1] Holme, Audun (2010). Geometry: Our Cultural Heritage. Springer. p. 210. ISBN 978-3-642-14440-0.

[2] Benitez, Julio (2007). "A Unified Proof of Ceva and Menelaus' Theorems Using Projective Geometry" (PDF). Journal for Geometry and Graphics. 11 (1): 39–44.

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