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So many angle bisectors

Source: 2021 China TST, Test 2, Day 1 P1

March 21, 2021
geometryincenterangle bisector

Problem Statement

A cyclic quadrilateral ABCDABCD has circumcircle Γ\Gamma, and AB+BC=AD+DCAB+BC=AD+DC. Let EE be the midpoint of arc BCDBCD, and F(≠C)F (\neq C) be the antipode of AA wrt Γ\Gamma. Let I,J,KI,J,K be the incenter of △ABC\triangle ABC, the AA-excenter of △ABC\triangle ABC, the incenter of △BCD\triangle BCD, respectively. Suppose that a point PP satisfies △BIC∼+△KPJ\triangle BIC \stackrel{+}{\sim} \triangle KPJ. Prove that EKEK and PFPF intersect on Γ.\Gamma.
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