OME 1993 Problem 5

Given a 4×4 grid of points, the points at two opposite corners are denoted $A$ and $D$ . We need to choose two other points $B$ and $C$ such that the six pairwise distances of these four points are all distinct. (a) How many such quadruples of points are there? (b) How many such quadruples of points are non-congruent? (c) If each point is assigned a pair of coordinates $(x_i,y_i)$ , prove that the sum of the expressions $|x_i-x_j |+|y_i-y_j|$ over all six pairs of points in a quadruple is constant.