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Circumscribed quadrilateral and incenters

Source: BMO SL 2023 G1

May 3, 2024
geometryincenter

Problem Statement

Let ABCDABCD be a circumscribed quadrilateral and let XX be the intersection point of its diagonals ACAC and BDBD. Let I1,I2,I3,I4I_1, I_2, I_3, I_4 be the incenters of △DXC\triangle DXC, △BXC\triangle BXC, △AXB\triangle AXB, and △DXA\triangle DXA, respectively. The circumcircle of △CI1I2\triangle CI_1I_2 intersects the sides CBCB and CDCD at points PP and QQ, respectively. The circumcircle of △AI3I4\triangle AI_3I_4 intersects the sides ABAB and ADAD at points MM and NN, respectively. Prove that AM+CQ=AN+CPAM+CQ=AN+CP
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