KöMaL problem A, 2018

Geometry Difficulty 7.0 National Olympiad, round 2 Prove it Hungary

In triangle $ABC$ with incenter $I$ , line $AI$ intersects the circumcircle of $ABC$ at $S\ne A$ . Let the reflection of $I$ with respect to $BC$ be $J$ , and suppose that line $SJ$ intersects the circumcircle of $ABC$ for the second time at point $P\ne S$ . Show that $AI=PI$ .

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