Menelaus's theorem

Statement

Let $ABC$ be a triangle in the Euclidean plane $\mathbb{R}^2$. For collinear points $X$, $Y$, $Z$ with $Z \neq Y$, let the signed ratio $\frac{XZ}{ZY}$ be the real number $\lambda$ with $Z - X = \lambda\,(Y - Z)$. Let $\ell$ be a line that does not pass through $A$, $B$ or $C$. Suppose that $\ell$ meets the lines $BC$, $CA$ and $AB$ at the points $D$, $E$ and $F$. Then $$\frac{AF}{FB} \cdot \frac{BD}{DC} \cdot \frac{CE}{EA} = -1.$$ Conversely, let $D$, $E$ and $F$ be points of the lines $BC$, $CA$ and $AB$, different from the vertices, with this product equal to $-1$. Then $D$, $E$ and $F$ are collinear.

Remarks

With unsigned lengths, the product is $1$. The converse of this weaker form is false in general. The formula differs from the formula of Ceva's theorem only in the sign. The line $\ell$ meets the inside of exactly two sides, or of no side. Thus one or three of the ratios are negative, and the product is negative.

History

The oldest surviving statement is in the Spherics of Menelaus of Alexandria. There, he uses the plane version as a lemma for a version on the sphere. Ptolemy used the theorem in the Almagest for problems of spherical astronomy.

Explanations

L1 human

A straight line that cuts across a triangle meets the three side lines in three points. These points cut the sides in certain ratios. Going round the triangle, the product of the three signed ratios is always $-1$.