Find the coordinates of C and D, such that ABCD is a rhombus. (see attached diagram)

I went ahead with simultaneous derived from distance formulas and perpendiular gradients and then solved that with Pythagoras theorem to end up with the points

C(10.5, 13.5) and D(5.5, 1.5).

Feeling very pleased with myself for solving it, i realised i had made an assumption that was incorrect. I assumed that ABCD was a square, and using Pythagoras, with AB = 13 units, then side AD = $\displaystyle \frac{13}{\sqrt2}$ and proceeded from there and it all worked fine.

Realising now the assumption was faulty, i have tried to do it without the property of ABCD being a square, and just cannot make any headway with it.

Any help greatly appreciated