# Reciprocal of Menelaus' Theorem

## Famous theorems in 2D Geometry

**Reciprocal of Menelaus' theorem. **Whether triangle *ABC* and the points *M*, *N*, *P* located on the lines *BC*,* CA*, respectively,* AB* different peaks *A*, *B*, *C*. If , the points *M*,* N*, *P* are **collinear**.

**Proof. **We have this figure

We suppose, hypothetically, that the points *M, N, P* are not collinear. Then there is the point *M*'∈ *BC*, so that *M', N, P* are collinear.

And then there is the relationship (according Menelaus's Theorem):

But we already have:

Further, we obtain

So though, we have **M' = M**. And, at the end, *M, N, P* are **collinear**.

**Keywords: **
geometry, Menelaus, reciprocal, mutual, triangle, collinear

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