BMO Round 2, 2025

Geometry Difficulty 7.0 National Olympiad, round 2 Prove it United Kingdom

In an acute-angled triangle $ABC$ with $AB < AC$ , the incentre is $I$ and the perpendicular bisector of $BC$ meets $BI$ at $P$ and $CI$ at $Q$ . The circles $BIQ$ and $CIP$ meet again at $X$ . The lines $AX$ and $BC$ meet at $D$ .

Prove that $D$ lies on the circle $AQP$ .

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