NL-BXMO-TST 2016 Problem 3

Let $\vartriangle ABC$ be a right-angled triangle with $\angle A = 90^o$ and circumcircle $\Gamma$ . The inscribed circle is tangent to $BC$ in point $D$ . Let $E$ be the midpoint of the arc $AB$ of $\Gamma$ not containing $C$ and let $F$ be the midpoint of the arc $AC$ of $\Gamma$ not containing $B$ . (a) Prove that $\vartriangle ABC \sim \vartriangle DEF$ . (b) Prove that $EF$ goes through the points of tangency of the incircle to $AB$ and $AC$ .