BG-MO 1967 Problem 3

It is given a right-angled triangle $ABC$ and its circumcircle $k$ . (a) prove that the radii of the circle $k_1$ tangent to the cathets of the triangle and to the circle $k$ is equal to the diameter of the incircle of the triangle ABC. (b) on the circle $k$ there may be found a point $M$ for which the sum $MA+MB+MC$ is as large as possible.