APMO 2026 - Asian Pacific Mathematics Olympiad Question 4

Let $ABCD$ be a quadrilateral with an incircle $\omega$ of center $I$. The diagonals $AC,BD$ intersect at $E$. Let $J$ be the incenter of triangle $ABD$. The extension of ray $EJ$ intersects $\omega$ at $P$. Prove that $PI\perp BD$.

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