Menelaus' Theorem

Famous theorems in 2D Geometry

Theorem. Whether the triangle ABC and the points M, N, P located on the lines BC, CA, AB respectively, different from A, B, C.
If the points M, N, P are collinear, then it has the relationship:

Menelaos formula in triangle

Proof. We have 2 possible situations. As in bellow figure:

Menelaos theorem

We will make the proof only for the first situation. The second one is similar.

We draw CS || BA, S ∈ MP.

In the triangle MNP with CS || BP, according to the Fundamental Theorem of similarity, we have:

And, again, inside the triangle CNS with CS || AP, we have:

We multiply member by member the above relations and we have:

Menelaos proof

and, finally:

Menelaos relation

q.e.d.

Keywords: Menelaos, theorem, proof, geometry, school, education, Menelaus

 

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