PlanetMath: proof of Menelaus' theorem

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proof of Menelaus' theorem

(Proof)

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

$\includegraphics{menelaus.eps}$

From this we follow (

and

being undircted):

$\displaystyle \frac{AZ}{ZB}$	$\displaystyle =$	$\displaystyle -\frac{h_1}{h_2}$
$\displaystyle \frac{BY}{YC}$	$\displaystyle =$	$\displaystyle \frac{h_2}{h_3}$
$\displaystyle \frac{CX}{XA}$	$\displaystyle =$	$\displaystyle \frac{h_3}{h_1}.$

Mulitplying all this we get:

$\displaystyle \frac{AZ}{ZB}\cdot\frac{BY}{YC}\cdot\frac{CX}{XA} = -\frac{h_1h_2h_3}{h_2h_3h_1} = -1.$

The second case is that the line connecting , and does not intersect any of the triangle's sides:

$\includegraphics{menelaus2.eps}$

In this case we get:

$\displaystyle \frac{AZ}{ZB}$	$\displaystyle =$	$\displaystyle -\frac{h_1}{h_2}$
$\displaystyle \frac{BY}{YC}$	$\displaystyle =$	$\displaystyle -\frac{h_2}{h_3}$
$\displaystyle \frac{CX}{XA}$	$\displaystyle =$	$\displaystyle -\frac{h_3}{h_1}.$