Let-I-be-the-incenter-of-ABC-It-is-known-that-for-every-point-M-AB-one-can-find-the-points-N-BC-and-P-AC-such-that-I-is-the-centroid-of-MNP-Prove-that-ABC-is-an-equilateral-triangle-

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Question Number 16873 by Tinkutara last updated on 27/Jun/17

Let I be the incenter of ΔABC. It is known that for every point M ∈ (AB), one can find the points N ∈ (BC) and P ∈ (AC) such that I is the centroid of ΔMNP. Prove that ABC is an equilateral triangle.

$$\mathrm{Let}\:{I}\:\mathrm{be}\:\mathrm{the}\:\mathrm{incenter}\:\mathrm{of}\:\Delta{ABC}.\:\mathrm{It}\:\mathrm{is} \\ $$$$\mathrm{known}\:\mathrm{that}\:\mathrm{for}\:\mathrm{every}\:\mathrm{point}\:{M}\:\in\:\left({AB}\right), \\ $$$$\mathrm{one}\:\mathrm{can}\:\mathrm{find}\:\mathrm{the}\:\mathrm{points}\:{N}\:\in\:\left({BC}\right)\:\mathrm{and} \\ $$$${P}\:\in\:\left({AC}\right)\:\mathrm{such}\:\mathrm{that}\:{I}\:\mathrm{is}\:\mathrm{the}\:\mathrm{centroid}\:\mathrm{of} \\ $$$$\Delta{MNP}.\:\mathrm{Prove}\:\mathrm{that}\:{ABC}\:\mathrm{is}\:\mathrm{an} \\ $$$$\mathrm{equilateral}\:\mathrm{triangle}. \\ $$

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Let-I-be-the-incenter-of-ABC-It-is-known-that-for-every-point-M-AB-one-can-find-the-points-N-BC-and-P-AC-such-that-I-is-the-centroid-of-MNP-Prove-that-ABC-is-an-equilateral-triangle-

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