BG-MO 1967 Problem 4

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.