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four points lie on one circle

Source: 2022 China Southeast Grade 10 P6

August 3, 2022
geometry

Problem Statement

HH is the orthocenter of △ABC\triangle ABC,the circle with center HH passes through AA,and it intersects with AC,ABAC,AB at two other points D,ED,E.The orthocenter of △ADE\triangle ADE is H′H',line AH′AH' intersects with DEDE at point FF.Point PP is inside the quadrilateral BCDEBCDE,so that △PDE∼△PBC\triangle PDE\sim\triangle PBC.Let point KK be the intersection of line HH′HH' and line PFPF.Prove that A,H,P,KA,H,P,K lie on one circle. https://i.ibb.co/mcyhxRM/graph.jpg
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