proof of Menelaus’ theorem

First we note that there are two different cases: Either the line connecting X, Y and Z intersects two sides of the triangleMathworldPlanetmath or none of them. So in the first case that it intersects two of the triangle’s sides we get the following picture:

From this we follow (h1, h2 and h3 being undircted):

AZZB = -h1h2
BYYC = h2h3
CXXA = h3h1.

Mulitplying all this we get:


The second case is that the line connecting X, Y and Z does not intersect any of the triangle’s sides:

In this case we get:

AZZB = -h1h2
BYYC = -h2h3
CXXA = -h3h1.

So multiplicationPlanetmathPlanetmath again yields Menelaus’ theoremMathworldPlanetmath.

Title proof of Menelaus’ theorem
Canonical name ProofOfMenelausTheorem
Date of creation 2013-03-22 12:46:46
Last modified on 2013-03-22 12:46:46
Owner mathwizard (128)
Last modified by mathwizard (128)
Numerical id 4
Author mathwizard (128)
Entry type Proof
Classification msc 51A05