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.