Question 3
Let $ABC$ be an acute triangle with fixed $B,C$ and variable $A$. Let $D$ be the midpoint of $BC$, and let $E,F$ be the perpendicular projections of $D$ onto $AB,AC$, respectively.
a) Let $O$ be the circumcenter of $ABC$. The line $EF$ meets $AO$ and $BC$ at $M$ and $N$, respectively. Prove that the circumcircle of $AMN$ passes through a fixed point.
b) The tangents at $E,F$ to the circumcircle of $AEF$ meet at $T$. Prove that $T$ belongs to a fixed line.
a) Let $O$ be the circumcenter of $ABC$. The line $EF$ meets $AO$ and $BC$ at $M$ and $N$, respectively. Prove that the circumcircle of $AMN$ passes through a fixed point.
b) The tangents at $E,F$ to the circumcircle of $AEF$ meet at $T$. Prove that $T$ belongs to a fixed line.