CN-TST 2021 Problem 7

A cyclic quadrilateral $ABCD$ has circumcircle $\Gamma$ , and $AB+BC=AD+DC$ . Let $E$ be the midpoint of arc $BCD$ , and $F (\neq C)$ be the antipode of $A$ wrt $\Gamma$ . Let $I,J,K$ be the incenter of $\triangle ABC$ , the $A$ -excenter of $\triangle ABC$ , the incenter of $\triangle BCD$ , respectively. Suppose that a point $P$ satisfies $\triangle BIC \stackrel{+}{\sim} \triangle KPJ$ . Prove that $EK$ and $PF$ intersect on $\Gamma.$