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.
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