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