---
tag: 07NSKDGS7S
kind: statement
role: theorem
name: Menelaus's theorem
msc: [51M04]
uses: [0CTNFGYMM2, 0MP9BCF6DP, 0AFNNTRTJE]
sources:
  - url: https://en.wikipedia.org/wiki/Menelaus%27s_theorem
    locator: lead section
  - url: https://en.wikipedia.org/wiki/Menelaus%27s_theorem
    locator: History
same_as:
  wikidata: Q193882
  wikipedia: Menelaus's theorem
---

## Statement

Let $ABC$ be a [[triangle|0CTNFGYMM2]] in the Euclidean plane $\mathbb{R}^2$. For [[collinear points|0AFNNTRTJE]] $X$, $Y$, $Z$ 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|0MP9BCF6DP]] 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|0KN93VDSXA]] 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 $-1$.