Problem
GEO-B2-M03-P023 Centres of All Orthogonal Circles
#23
★★★★★ Level 5 of 5
Several circles are orthogonal to two fixed circles \(\omega_1\) and \(\omega_2\). Prove that the centres of all these circles lie on one line.
The centre of each such circle lies on the radical axis of \(\omega_1\) and \(\omega_2\).
Let \(X\) be the centre of one of the circles orthogonal to \(\omega_1\) and \(\omega_2\). By the problem about the centre of an orthogonal circle, \(X\) lies on the radical axis of \(\omega_1\) and \(\omega_2\). The same is true for the centre of any other such circle. Therefore all these centres lie on one line: the radical axis of the fixed circles.
A strong collinearity result without angles: all points have the same radical reason.