Problem

A cube ABCDA'B'C'D' is given.
(a) Find the locus of all midpoints of segments XY, where X is an point on segment AC and Y is any point on segment B'D'.
(b) Find the locus of all points Z on segments XY such that ZY=2XZ.

Solution

(a) The locus of the points is the square EFGH where these four points are the centers of of the faces ABB'A', BCC'B', CDD'C', and DAA'D', respectively.

(b) The locus of the points is the rectangle IJKL where these points are on AB', CB', CD', and AD' at a distance of AA'/3 with respect to plane ABCD.