Problem

Source: VMO 2019

Tags: geometry, concurrency, Concyclic



Let $ABC$ be an acute, nonisosceles triangle with inscribe in a circle $(O)$ and has orthocenter $H$. Denote $M,N,P$ as the midpoints of sides $BC,CA,AB$ and $D,E,F$ as the feet of the altitudes from vertices $A,B,C$ of triangle $ABC$. Let $K$ as the reflection of $H$ through $BC$. Two lines $DE,MP$ meet at $X$; two lines $DF,MN$ meet at $Y$. a) The line $XY$ cut the minor arc $BC$ of $(O)$ at $Z$. Prove that $K,Z,E,F$ are concyclic. b) Two lines $KE,KF$ cuts $(O)$ second time at $S,T$. Prove that $BS,CT,XY$ are concurrent.