MathDB

9

Part of 1994 Moldova Team Selection Test

Problems(1)

$AB+BC=AD+CD$, $BC+CA=BD+AD$, $CA+AB=CD+BD$

Source: Moldova TST 1994

8/8/2023
Let OO{} be the center of the circumscribed sphere of the tetrahedron ABCDABCD. Let L,M,NL,M,N respectively be the midpoints of the segments BC,CA,ABBC,CA,AB. It is known that AB+BC=AD+CDAB+BC=AD+CD, BC+CA=BD+ADBC+CA=BD+AD, CA+AB=CD+BDCA+AB=CD+BD. Prove that ∠LOM=∠MON=∠NOL\angle LOM=\angle MON=\angle NOL. Find their value.
geometry3D geometryspheretetrahedron