Menelaus’ theorem

If the points $X$, $Y$ and $Z$ are on the sides of a triangle $ABC$ (including their prolongations), collinear and do not coincide with any of the points $A$, $B$ and $C$, then the equation

$$\frac{AZ}{ZB}\cdot \frac{BY}{YC}\cdot \frac{CX}{XA}=-1$$

holds (all segments are directed line segments). The converse of this theorem also holds (thus: three points on the prolongations of the triangle’s sides are collinear if the above equation holds).

Title	Menelaus’ theorem
Canonical name	MenelausTheorem
Date of creation	2013-03-22 12:46:20
Last modified on	2013-03-22 12:46:20
Owner	mathwizard (128)
Last modified by	mathwizard (128)
Numerical id	7
Author	mathwizard (128)
Entry type	Theorem
Classification	msc 51A05
Related topic	CevasTheorem
Related topic	TrigonometricVersionOfCevasTheorem
Related topic	Collinear