Question 58
Let $ABC$ be an acute triangle with circumcircle $k$ and incenter $I$. The line through $I$ perpendicular to $CI$ meets side $BC$ at $U$ and $k$ at $V$, where $V,A$ lie on opposite sides of $BC$. The line through $U$ parallel to $AI$ meets $AV$ at $X$.
Prove: If $XI$ and $AI$ are perpendicular, then $XI$ meets side $AC$ at its midpoint $M$.
Prove: If $XI$ and $AI$ are perpendicular, then $XI$ meets side $AC$ at its midpoint $M$.