Trigonometry/Circles and Triangles/Menelaus' theorem

Menelaus' theorem is due to the Greek philosopher Menelaus of Alexandria.

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Let ABC be any triangle. Let a straight line cross BC at D, AC at E and AB at F (extending one side as necessary). Then

$\displaystyle {\frac {BD}{DC}}{\frac {CE}{EA}}{\frac {AF}{FB}}=-1$.

The product is negative because one side has to be produced and so one of the segments must be treated as of negative length.

Conversely, if three points on the three sides of a triangle satisfy the above relationship, they must be collinear.