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Geometry with rectangle

Source: 239-School Open Olympiad 2022, Junior League P6

April 19, 2023
geometrycircles

Problem Statement

On the side BCBC of the rectangle ABCDABCD, a point PP{} is marked so that ∠APD=90∘\angle APD = 90^\circ. On the straight line ADAD, points QQ{} and RR{} are selected outside the segment ADAD such that AQ=BPAQ = BP and CP=DRCP = DR. The circle ω\omega passes through the points Q,DQ, D and the circumcenter of the triangle PDQPDQ. The circle γ\gamma passes through the points A,RA, R and the circumcenter of the triangle APRAPR. Prove that the radius of one of the circles touching the line ADAD and the circles ω\omega and γ\gamma is 2AB2AB.
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