IMO 1983 LL AUS3

Origin: AUS

Problem

(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 H4 of ∆ABC. Prove that this altitude DH4 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 H1 of ∆BCD, then prove that all four altitudes are concurrent and each one passes through the orthocenter of the respective triangle.