Question 179
Given a triangle $ABC$ with fixed vertices $B$, $C$; point $A$ moves such that triangle $ABC$ is acute. Let $D$ be the midpoint of $BC$ and $E$, $F$ be the projections of $D$ to $AB$, $AC$ respectively.
a) Let $O$ be the circumcenter of triangle $ABC$. $EF$ meets $AO$ and $BC$ at $M$, $N$ respectively. Prove that the circumcircle of triangle $AMN$ passes through a fixed point.
b) Suppose that the tangents of the circumcircle of triangle $AEF$ at $E$, $F$ intersect each other at $T$. Prove that $T$ lies on a fixed line.
a) Let $O$ be the circumcenter of triangle $ABC$. $EF$ meets $AO$ and $BC$ at $M$, $N$ respectively. Prove that the circumcircle of triangle $AMN$ passes through a fixed point.
b) Suppose that the tangents of the circumcircle of triangle $AEF$ at $E$, $F$ intersect each other at $T$. Prove that $T$ lies on a fixed line.