Problem
GEO-B3-M01-P020 Centers of Orthogonal Circles
Two nonconcentric circles \(\omega_1\) and \(\omega_2\) are given. Prove that the centers of all circles orthogonal to both given circles lie on the radical axis of \(\omega_1\) and \(\omega_2\).
Hint 1. Let the required circle have center \(X\) and radius \(r\).
Hint 2. Write the orthogonality condition using the distance between centers.
E. Full solution. Let the centers of the given circles be \(O_1,O_2\), their radii \(R_1,R_2\), and let a circle centered at \(X\) with radius \(r\) be orthogonal to both. Then \(XO_1^2=R_1^2+r^2\) and \(XO_2^2=R_2^2+r^2\). Subtracting gives \(XO_1^2-R_1^2=XO_2^2-R_2^2\). This means that the powers of \(X\) with respect to \(\omega_1\) and \(\omega_2\) are equal. Therefore \(X\) lies on their radical axis.
Although the problem is solved by powers, it is needed for inversion: such circles are often chosen as circles of inversion.