ILL 1983 Problem 3

(a) Given a tetrahedron $ABCD$ and its four altitudes (i.e., lines through each vertex, perpendicular to the opposite face), assume that the altitude dropped from $D$ passes through the orthocenter $H_4$ of $\triangle ABC$ . Prove that this altitude $DH_4$ intersects all the other three altitudes.(b) If we further know that a second altitude, say the one from vertex A to the face $BCD$ , also passes through the orthocenter $H_1$ of $\triangle BCD$ , then prove that all four altitudes are concurrent and each one passes through the orthocenter of the respective triangle.