CMO 1982 Problem 5

The altitudes of a tetrahedron $ABCD$ are extended externally to points $A'$ , $B'$ , $C'$ , and $D'$ , where $AA' = k/h_a$ , $BB' = k/h_b$ , $CC' = k/h_c$ , and $DD' = k/h_d$ . Here, $k$ is a constant and $h_a$ denotes the length of the altitude of $ABCD$ from vertex $A$ , etc. Prove that the centroid of tetrahedron $A'B'C'D'$ coincides with the centroid of $ABCD$ .