IrMO 2011 Problem 4

The incircle $\mathcal{C}_1$ of triangle $ABC$ touches the sides $AB$ and $AC$ at the points $D$ and $E$ , respectively. The incircle $\mathcal{C}_2$ of the triangle $ADE$ touches the sides $AB$ and $AC$ at the points $P$ and $Q$ , and intersects the circle $\mathcal{C}_1$ at the points $M$ and $n$ . Prove that

(a) the center of the circle $\mathcal{C}_2$ lies on the circle $\mathcal{C}_1$ .

(b) the four points $M,N,P,Q$ in appropriate order form a rectangle if and only if twice the radius of $\mathcal{C}_1$ is three times the radius of $\mathcal{C}_2$ .