Finding the coordinates of the third point in a triangle using simultaneous equations

Question

I have 3 circles A,B & C that touch each other at tangents. The centre points of these 3 circles are to be joined to create a triangle. I know the coordinates of 2 of the circles centre points A(x_a, y_a) and B(x_b, y_b) and I know the radii of all 3 circles r_a, r_b & r_c. I need to calculate C(x_c, y_c) that is the centre point of circle C using simultaneous equations. Step by step would be appreciated but just the answer is also fine.

I know this can be done easier using the Law of Cosines but I'm not allowed to use this method in this situation.