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IGO 2022 advanced/free P4

Source: Iranian Geometry Olympiad 2022 P4 Advanced, Free

December 13, 2022
geometry

Problem Statement

Let ABCDABCD be a trapezoid with AB∥CDAB\parallel CD. Its diagonals intersect at a point PP. The line passing through PP parallel to ABAB intersects ADAD and BCBC at QQ and RR, respectively. Exterior angle bisectors of angles DBADBA, DCADCA intersect at XX. Let SS be the foot of XX onto BCBC. Prove that if quadrilaterals ABPQABPQ, CDQPCDQP are circumcribed, then PR=PSPR=PS.
Proposed by Dominik Burek, Poland
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