BG-MO 1967

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.

Outside of the plane of the triangle $ABC$ is given point $D$ . (a) prove that if the segment $DA$ is perpendicular to the plane $ABC$ then orthogonal projection of the orthocenter of the triangle $ABC$ on the plane $BCD$ coincides with the orthocenter of the triangle $BCD$ . (b) for all tetrahedrons $ABCD$ with base, the triangle $ABC$ with smallest of the four heights that from the vertex $D$ , find the locus of the foot of that height.