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geometry in P4?

Source: KMO 2023 P4

November 4, 2023
geometrycircumcircle

Problem Statement

Pentagon ABCDEABCDE is inscribed in circle Ω\Omega. Line ADAD meets CECE at FF, and P(≠E,F)P (\neq E, F) is a point on segment EFEF. The circumcircle of triangle AFPAFP meets Ω\Omega at Q(≠A)Q(\neq A) and ACAC at R(≠A)R(\neq A). Line ADAD meets BQBQ at SS, and the circumcircle of triangle DESDES meets line BQ,BDBQ, BD at T(≠S),U(≠D)T(\neq S), U(\neq D), respectively. Prove that if F,P,T,SF, P, T, S are concyclic, then P,T,R,UP, T, R, U are concyclic.
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