I need to find the center of the smallest circle which is orthogonal to two other circles. I know that the center of circles orthogonal to two other circles will lie on the radical axis of those two circles.
My book's solution for this question, states that the required center lies on the intersection of the radical axis of the two given points and the line joining their centers.
I do know that it lies on the radical axis, but does the center of the (smallest) circle also lie on the line joining the centers? This is not stated as a fact and isn't particularly obvious to me, so can someone help me with a proof? I tried to prove this but wasn't able to get the conditions for the circle to be "smallest".
I also know that the common chord of two intersecting circles is the same as the radical axis, and the common chord is bisected by the line joining the centers.
Using these, I was not able to come up with a proof, any help would be appreciated greatly