Question 178
Let $\triangle ABC$ be a right triangle with $\angle A = 90^\circ$ and circumcircle $\Gamma$. The incircle touches $BC$ at a point $D$. Let $E$ be the midpoint of the arc $AB$ of $\Gamma$ on which $C$ does not lie and let $F$ be the midpoint of the arc $AC$ of $\Gamma$ on which $B$ does not lie.
a) Prove that $\triangle ABC \sim \triangle DEF$.
b) Prove that $EF$ passes through the points where the incircle touches $AB$ and $AC$.
a) Prove that $\triangle ABC \sim \triangle DEF$.
b) Prove that $EF$ passes through the points where the incircle touches $AB$ and $AC$.