Problem

Source: Ukraine 1997 grade 11

Tags: geometry, circumcircle, parallelogram, modular arithmetic, 3D geometry, sphere, geometry proposed



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.