Some possibilities that you could do other things with as well (start off with a couple of concrete cases and extend to the general case):
Subsets of the complex plane defined by:
|z - z1| + |z - z2| = k and |z - z1| - |z - z2| = k
|z - z1| = k |z - z2| (where k > 0)
arg(z - z1) - arg(z - z2) = theta.
The second and third can also incorporate vectors and circle geometry theorems.