This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

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Found problems: 1

2019 Danube Mathematical Competition, 4
Tags: geometry , circumcircle , orthocentre
Let $ APD $ be an acute-angled triangle and let $ B,C $ be two points on the segments (excluding their endpoints) $ AP,PD, $ respectively. The diagonals of $ ABCD $ meet at $ Q. $ Denote by $ H_1,H_2 $ the orthocenters of $ APD,BPC, $ respectively. The circumcircles of $ ABQ $ and $ CDQ $ intersect at $ X\neq Q, $ and the circumcircles of $ ADQ,BCQ $ meet at $ Y\neq Q. $ Prove that if the line $ H_1H_2 $ passes through $ X, $ then it also passes through $ Y. $