Obtuse Angle

Prove that among any 9 points in the space (3D), there are three which form an obtuse angle.


Let the points be labeled A1, A2, ... , A9, and P be their convex hull. If we assume that all angles among the points are non-abtuse, then the interiors of the bodies P+A1, P+A2, ... , P+A9 should be all disjoint. That is because for every Ai and Aj, P must be bounded between the planes passing through Ai, Aj, and orthogonal to the segment AiAj. However, all of these 9 bodies have the same volume and are contained in the body P+P, which has 8 times larger volume. This is a contradiction, and therefore our assumption is wrong.