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A,B,C,D,X concyclic if ABCD cyclic, PC = AD

Source: 2021 Chile National Olympiad level 2 p4

October 22, 2022
geometryConcycliccyclic quadrilateral

Problem Statement

Consider quadrilateral ABCDABCD with ∣DC∣>∣AD∣|DC| > |AD|. Let PP be a point on DCDC such that PC=ADPC = AD and let QQ be the midpoint of DPDP. Let L1L_1 be the line perpendicular on DCDC passing through QQ and let L2L_2 be the bisector of the angle ∠ABC \angle ABC. Let us call X=L1∩L2X = L_1 \cap L_2. Show that if quadrilateral is cyclic then XX lies on the circumcircle of ABCD.ABCD. https://cdn.artofproblemsolving.com/attachments/f/6/3ebfce8a7fd2a0ece9f09065608141006893d2.png
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