Theorem of Menelaus

Menelaus's theorem, named for Menelaus of Alexandria, is a proposition about triangles in plane geometry. Given a triangle ABC, and a transversal line that crosses BC, AC, and AB at points D, E, and F respectively, with D, E, and F distinct from A, B, and C, then

or simply

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 D, E, and F are collinear. The converse is often included as part of the theorem.

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

A standard proof is as follows:

First, the sign of the left-hand side will be negative since either all three of the ratios are negative, the case where the line DEF misses the triangle (lower diagram), or one is negative and the other two are positive, the case where DEF crosses two sides of the triangle. (See Pasch's axiom.)

To check the magnitude, construct perpendiculars from A, B, and C to the line DEF and let their lengths be a, b, and c respectively. Then by similar triangles it follows that |AF/FB| = |a/b|, |BD/DC| = |b/c|, and |CE/EA| = c/a. So

For a simpler, if less symmetrical way to check the magnitude, draw CK parallel to AB where DEF meets CK at K. Then by similar triangles

and the result follows by eliminating CK from these equations.

The converse follows as a corollary. Let D, E, and F be given on the lines BC, AC, and AB so that the equation holds. Let F′ be the point where DE crosses AB. Then by the theorem, the equation also holds for D, E, and F′. Comparing the two,

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