GeoProof — Menelaus' Theorem explorer

(AF / FB) · (BD / DC) · (CE / EA) = 1

Drag the two square handles to move the transversal, or drag a vertex. One of D, E, F always lands on a side's extension — that's unavoidable for a straight cut. If a point runs off the canvas, a colored arrow on the edge points toward it, so you can drag the line to bring it back.

Menelaus' Theorem

Draw a straight line (a transversal) that crosses the three side-lines of triangle ABC — meeting line BC at D, line CA at E, and line AB at F. Menelaus' theorem says the three division ratios multiply to one:

(AF / FB) · (BD / DC) · (CE / EA) = 1

Here the lengths are taken as positive. With signed ratios the product is −1, which encodes a key fact: a straight line must cut either two of the sides and the extension of the third, or all three extensions — never all three sides between the vertices.

The twin of Ceva

Ceva's theorem uses the very same product but equals +1 (signed), and it is about three cevians being concurrent. Menelaus equals −1 and is about three points being collinear. Concurrence and collinearity are dual ideas, and these two theorems are their algebraic statements.

What it's used for

Menelaus is a standard tool for proving points collinear and for chasing ratios along lines — it underlies many results in projective geometry and competition problems.

Converse

The converse holds too: if three points on the side-lines give a product of 1 (signed −1), then they are collinear. So the equation is a test for collinearity.

A bit of history

It is named for Menelaus of Alexandria (c. 70–140 CE), who stated a version for great-circle arcs on a sphere in his Sphaerica; the planar version is the one shown here.