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Cyclic quadrilateral and a tangent

Source: IMOC 2023 G6

September 9, 2023
geometry

Problem Statement

Triangle ABCABC has circumcenter OO. DD is the foot from AA to BCBC, and PP is apoint on ADAD. The feet from PP to CA,ABCA, AB are E,FE, F, respectively, and the foot from DD to EFEF is TT. AOAO meets (ABC)(ABC) again at A′A'. A′DA'D meets (ABC)(ABC) again at RR. If QQ is a point on AOAO satisfying ∠ABP=∠QBC\angle ABP = \angle QBC, prove that D,P,T,RD, P, T, R lie on acircle and DQDQ is tangent to it.
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