Question 46
The incircle $C_1$ of triangle $ABC$ touches the sides $AB$ and $AC$ at the points $D$ and $E$, respectively. The incircle $C_2$ of the triangle $ADE$ touches the sides $AB$ and $AC$ at the points $P$ and $Q$, and intersects the circle $C_1$ at the points $M$ and $N$. Prove that
(a) the centre of the circle $C_2$ lies on the circle $C_1$;
(b) the four points $M, N, P, Q$ in appropriate order form a rectangle if and only if twice the radius of $C_1$ is three times the radius of $C_2$.
(a) the centre of the circle $C_2$ lies on the circle $C_1$;
(b) the four points $M, N, P, Q$ in appropriate order form a rectangle if and only if twice the radius of $C_1$ is three times the radius of $C_2$.