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concurrency wanted, starting with a tangential ABCD

Source: 2019 RMM Shortlist G5

June 19, 2020
geometryperimeter

Problem Statement

A quadrilateral ABCDABCD is circumscribed about a circle with center II. A point P≠IP \ne I is chosen inside ABCDABCD so that the triangles PAB,PBC,PCD,PAB, PBC, PCD, and PDAPDA have equal perimeters. A circle Γ\Gamma centered at PP meets the rays PA,PB,PCPA, PB, PC, and PDPD at A1,B1,C1A_1, B_1, C_1, and D1D_1, respectively. Prove that the lines PI,A1C1PI, A_1C_1, and B1D1B_1D_1 are concurrent.
Ankan Bhattacharya, USA
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