MathDB

4

Part of 2019 Danube Mathematical Competition

Problems(2)

Inscriptible quadrilateral; angle chasing

Source: 2019 Danube

10/29/2019
Let ABCD ABCD be a cyclic quadrilateral,M M midpoint of AC AC and N N midpoint of BD. BD. If ∠AMB=∠AMD, \angle AMB =\angle AMD, prove that ∠ANB=∠BNC. \angle ANB =\angle BNC.
geometrycyclic quadrilateral
Hard pure geometry

Source: 2019 Danube

10/29/2019
Let APD APD be an acute-angled triangle and let B,C B,C be two points on the segments (excluding their endpoints) AP,PD, AP,PD, respectively. The diagonals of ABCD ABCD meet at Q. Q. Denote by H1,H2 H_1,H_2 the orthocenters of APD,BPC, APD,BPC, respectively. The circumcircles of ABQ ABQ and CDQ CDQ intersect at X≠Q, X\neq Q, and the circumcircles of ADQ,BCQ ADQ,BCQ meet at Y≠Q. Y\neq Q. Prove that if the line H1H2 H_1H_2 passes through X, X, then it also passes through Y. Y.
geometrycircumcircleOrthocentre