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Segments of equal length in a circumscribed rectangle

Source: Italy MO 2024 P4

May 15, 2024
geometry proposedgeometryrectanglecircumcircle

Problem Statement

Let ABCDABCD be a rectangle with AB<BCAB<BC and circumcircle Γ\Gamma. Let PP be a point on the arc BCBC (not containing AA) and let QQ be a point on the arc CDCD (not containing AA) such that BP=CQBP=CQ.
The circle with diameter AQAQ intersects APAP again in SS. The perpendicular to AQAQ through BB intersects APAP in XX. (a) Show that XS=PSXS=PS. (b) Show that AX=DQAX=DQ.
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