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common incircle of two triangles

Source: Croatia TST 2016

April 27, 2016
geometrycircumcircleincirclegeometry proposed

Problem Statement

Let PP be a point inside a triangle ABCABC such that AP+BPAB=BP+CPBC=CP+APCA. \frac{AP + BP}{AB} = \frac{BP + CP}{BC} = \frac{CP + AP}{CA} . Lines APAP, BPBP, CPCP intersect the circumcircle of triangle ABCABC again in A′A', B′B', C′C'. Prove that the triangles ABCABC and A′B′C′A'B'C' have a common incircle.
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