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  Menelaus theorem formula, the points M, N, P are collinear.

 

Proof. We have this figure

Menelaus theorem mutual

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):

Menelaus theorem

But we already have:

Menelaus theorem reciproc

Further, we obtain

menelaus mutuall reciprocal

Menelaus reciproc

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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