Suppose that we four random variables, A, B, C, D that have the same standard deviation. Here, A, B, C, and D are chosen randomly and independently of each other. Define E = -A-B+kC+(2-k)D and F = -A-B+(2-k)C+kD, where k is a real constant. Find the set of values of k such that the variables E and F have zero correlation.

Upon experimenting with large randomly-generated data sets of Microsoft Excel, I am led to believe that the possible values of k are 1 +- sqrt(2), but I have no idea how to prove that these are the possible values of k.