Geometry • Topic 22

Menelaus' Theorem

While Ceva's theorem deals with concurrency, Menelaus' Theorem deals with collinearity. It determines when three points on the sides (or extensions) of a triangle lie on a straight line.

Statement

Theorem (Menelaus' Theorem)

Let points

D, E, F

lie on lines

BC, C A, A B

respectively (where at least one is on an extension). Points

D, E, F

are collinear if and only if:

\frac{A F}{FB} \cdot \frac{B D}{D C} \cdot \frac{CE}{E A} = - 1

(Note: We use signed lengths. If the product of lengths is just 1, we must check that the line cuts the triangle externally an even number of times).

Comparison with Ceva

Ceva: Product = $+ 1$ (Concurrency).
Menelaus: Product = $- 1$ (Collinearity).

Practice Problems

Exercise (Problem 1)

△ A BC

D

is the midpoint of

BC

E

is on

A C

such that

A E = 2 EC

. Line

D E

intersects

A B

extension at

F

. Find

A F / FB

Exercise (Problem 2)

Prove that the external bisectors of two angles of a triangle and the internal bisector of the third angle are collinear points on the opposite sides.

Exercise (Problem 3)

(Desargues' Theorem Step) Use Menelaus to prove properties of perspective triangles.