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0672 geometry 6th edition Round 7 p2

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May 3, 2021
geometry6th edition

Problem Statement

Let ABCDABCD be a cyclic quadrilateral. Let M,NM, N be the midpoints of the diagonals ACAC and BDBD and let PP be the midpoint of MNMN. Let A′,B′,C′,D′A',B',C',D' be the intersections of the rays APAP, BPBP, CPCP and DPDP respectively with the circumcircle of the quadrilateral ABCDABCD.
Find, with proof, the value of the sum σ=APPA′+BPPB′+CPPC′+DPPD′. \sigma = \frac{ AP}{PA'} + \frac{BP}{PB'} + \frac{CP}{PC'} + \frac{DP}{PD'} .
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