UA-UMO 1997 Problem 118

On the edges $AB,BC,CD,DA$ of a parallelepiped $ABCDA_1 B_1 C_1 D_1$ points $K,L,M,N$ are selected, respectively. Prove that the circumcenters of the tetrahedra $A_1 AKN, B_1 BKL, C_1 CLM, D_1 DMN$ are vertices of a parallelogram.