Question 6
Let $ABC$ be an acute triangle, let $M$ be the midpoint of $BC$, and let $H$ be the foot of the altitude from $B$. Let $Q$ be the circumcenter of triangle $ABM$, and let $X$ be the intersection of the altitude $BH$ with the perpendicular bisector of $BC$.
Prove that the following two statements are equivalent:
(i) The circumcircle of triangle $ACM$, the circumcircle of triangle $AXH$, and the line $CQ$ pass through a common point.
(ii) The lines $BQ$ and $CQ$ are perpendicular.
Prove that the following two statements are equivalent:
(i) The circumcircle of triangle $ACM$, the circumcircle of triangle $AXH$, and the line $CQ$ pass through a common point.
(ii) The lines $BQ$ and $CQ$ are perpendicular.