KMO 2004 Problem 5

$A, B, C$ , and $D$ are the four different points on the circle $O$ in the order. Let the centre of the scribed circle of triangle $ABC$ , which is tangent to $BC$ , be $O_1$ . Let the centre of the scribed circle of triangle $ACD$ , which is tangent to $CD$ , be $O_2$ .

(1) Show that the circumcentre of triangle $ABO_1$ is on the circle $O$ .

(2) Show that the circumcircle of triangle $CO_1O_2$ always pass through a fixed point on the circle $O$ , when $C$ is moving along arc $BD$ .