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Geometry

Source: Moldova TST 2014, Third Day, Problem 3

March 31, 2014
geometrycircumcircletrigonometrygeometry proposed

Problem Statement

Let △ABC\triangle ABC be a triangle with ∠A\angle A-acute. Let PP be a point inside △ABC\triangle ABC such that ∠BAP=∠ACP\angle BAP = \angle ACP and ∠CAP=∠ABP\angle CAP =\angle ABP. Let M,NM, N be the centers of the incircle of △ABP\triangle ABP and △ACP\triangle ACP, and RR the radius of the circumscribed circle of △AMN\triangle AMN. Prove that 1R=1AB+1AC+1AP.\displaystyle \frac{1}{R}=\frac{1}{AB}+\frac{1}{AC}+\frac{1}{AP}.
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