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concurrent and concyclic wanted, symmtetrics wrt lines, cyclic ABCD

Source: 2011 XIV All-Ukrainian Tournament of Young Mathematicians, Qualifying p14

May 18, 2021
geometryconcurrencyConcyclicsymmetrycyclic quadrilateralUkrainian TYM

Problem Statement

Given a quadrilateral ABCDABCD, inscribed in a circle ω\omega such that AB=ADAB=AD and CB=CDCB=CD . Take the point P∈ωP \in \omega. Let the vertices of the quadrilateral Q1Q2Q3Q4Q_1Q_2Q_3Q_4 be symmetric to the point P wrt the lines ABAB, BCBC, CDCD, and DADA, respectively. a) Prove that the points symmetric to the point PP wrt lines Q1Q22,Q2Q3,Q3Q4Q_1Q_22, Q_2Q_3, Q_3Q_4 and Q4Q1Q_4Q_1, lie on one line. b) Prove that when the point PP moves in a circle ω\omega, then all such lines pass through one common point.
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