1. Two edges PQ, RS of a tetrahedron PQRS are perpendicular; show that the distance between the mid-points of PS and QR is equal to the distance between the mid-points of PR and QS.

Answer 1

You can set up the vertices PQRS as points on the x,y,z axes such that

P = (0,0,c)
Q = (0,b,0)
R = (0,0,0)
S = (a,0,0)

Then the midpoints are:

PS: (a/2,0,c/2)
QR: (0,b/2,0)

PR: (0,0,c/2)
QS: (a/2,b/2,0)

You can easily show that the lengths of those segments are equal.

There's probably some sly similar-triangles way to show it as well, but I'm too lazy just now.