Problem
GEO-B2-M03-P016 Two Orthogonal Circles
#16
★★★★☆ Level 4 of 5
Two distinct circles \(\gamma_1\) and \(\gamma_2\) are orthogonal to both circles \(\omega_1\) and \(\omega_2\). Prove that the centres of \(\gamma_1\) and \(\gamma_2\) lie on the radical axis of \(\omega_1\) and \(\omega_2\).
Apply the previous fact to each of the circles \(\gamma_1\) and \(\gamma_2\).
Let the centres of \(\gamma_1\) and \(\gamma_2\) be \(X_1\) and \(X_2\). Since \(\gamma_1\) is orthogonal to both given circles, by the previous fact \(X_1\) lies on their radical axis. Similarly, \(X_2\) lies on the same radical axis. Therefore both centres lie on one radical axis.
A good problem on reusing a lemma rather than computing.