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angles of lines wanted, heart by 3 semicircles constructed

Source: Iranian Geometry Olympiad 2021 IGO Elementary p3

January 25, 2022
geometryperimeteranglessemicircle

Problem Statement

As shown in the following figure, a heart is a shape consist of three semicircles with diameters ABAB, BCBC and ACAC such that BB is midpoint of the segment ACAC. A heart ω\omega is given. Call a pair (P,P′)(P, P') bisector if PP and P′P' lie on ω\omega and bisect its perimeter. Let (P,P′)(P, P') and (Q,Q′)(Q,Q') be bisector pairs. Tangents at points P,P′,QP, P', Q, and Q′Q' to ω\omega construct a convex quadrilateral XYZTXYZT. If the quadrilateral XYZTXYZT is inscribed in a circle, find the angle between lines PP′PP' and QQ′QQ'. https://cdn.artofproblemsolving.com/attachments/3/c/8216889594bbb504372d8cddfac73b9f56e74c.png
Proposed by Mahdi Etesamifard - Iran
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