For now, I'll work with your two example points. Using the distance formula and knowing that the two points A and B must be equidistant from C, we get:
\(\displaystyle \sqrt{(2 - X_C)^2 + (1 - Y_C)^2} = \sqrt{(3 - X_C)^2 + (2 - Y_C)^2}\)
If we expand everything out, we get:
\(\displaystyle \sqrt{(X_C)^2 - 4X_C + (Y_C)^2 - 2Y_C + 5} = \sqrt{(X_C)^2 - 6X_C + (Y_C)^2 - 4Y_C + 13}\)
Now, for any positive x and y, we have the property that \(\displaystyle \sqrt{x} = \sqrt{y} \implies x = y\). Since distance is always positive, we can just "delete" the square roots:
\(\displaystyle (X_C)^2 - 4X_C + (Y_C)^2 - 2Y_C + 5 = (X_C)^2 - 6X_C + (Y_C)^2 - 4Y_C + 13\)
Collecting all the variable terms on one side and the constants on the other, we see that the squared terms cancel out and we're left with:
\(\displaystyle 2X_C + 2Y_C = 8 \implies X_C + Y_C = 4\)
It should be easy enough for you to do the last step and figure out where this line intersects your circles. In fact, because you know the circles also intersect with each other at these points, it would be sufficient to find where the line intersects with one of them. And finally, what would happen if we left the coordinates of A and B as unknown? Well, we'd have:
\(\displaystyle (X_A - X_C)^2 + (Y_A - Y_C)^2 = (X_B - X_C)^2 + (Y_B - Y_C)^2\)
Try continuing from here and see what you can come up with.