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Bicentric quads formed by cevians

Source: India TST 2018, D4 P1

July 18, 2018
geometrycircumcircle

Problem Statement

Let ABCABC be a triangle and AD,BE,CFAD,BE,CF be cevians concurrent at a point PP. Suppose each of the quadrilaterals PDCE,PEAFPDCE,PEAF and PFBDPFBD has both circumcircle and incircle. Prove that ABCABC is equilateral and PP coincides with the center of the triangle.
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