IT-CESENATICO 1991 Problem 1

For every triangle $ABC$ inscribed in a circle $\Gamma$ , let $A',B',C'$ be the intersections of the bisectors of the angles at $A,B,C$ with $\Gamma$ . Consider the triangle $A'B'C'$ . (a) Do triangles $A'B'C'$ go over all possible triangles inscribed in $\Gamma$ as $\vartriangle ABC$ varies? If not, what are the constraints? (b) Prove that the angle bisectors of $\vartriangle ABC$ are the altitudes of $\vartriangle A',B',C'$ .