Problem
GEO-B2-M10-P008 Three Radical Axes
Circles \(\omega_1,\omega_2,\omega_3\) meet pairwise: \(\omega_1\) and \(\omega_2\) at \(A,B\), \(\omega_2\) and \(\omega_3\) at \(C,D\), and \(\omega_3\) and \(\omega_1\) at \(E,F\). Suppose lines \(AB\) and \(CD\) meet at \(X\). Prove that \(X\) lies on line \(EF\).
Point \(X\) has equal powers with respect to the first two circles and also the second and third.
Since \(X\in AB\), it lies on the radical axis of \(\omega_1,\omega_2\), so \(\operatorname{Pow}_{\omega_1}(X)=\operatorname{Pow}_{\omega_2}(X)\). Since \(X\in CD\), we get \(\operatorname{Pow}_{\omega_2}(X)=\operatorname{Pow}_{\omega_3}(X)\). Therefore \(\operatorname{Pow}_{\omega_1}(X)=\operatorname{Pow}_{\omega_3}(X)\), so \(X\) lies on the radical axis of \(\omega_1,\omega_3\), which is line \(EF\).
This is the basic form of the radical center theorem.