IMO 1977 LL CZS8

Origin: CZS

Problem

A hexahedron ABCDE is made of two regular congruent tetra- hedra ABCD and ABCE. Prove that there exists only one isometry Z that maps points A, B, C, D, E onto B, C, A, E, D, respectively. Find all points X on the surface of hexahedron whose distance from Z(X) is minimal.

Solution

There is exactly one point satisfying the given condition on each face of the hexahedron. Namely, on the face ABD it is the point that divides the median from D in the ratio 32 : 3.