Problem
GEO-B2-M03-P013 The Third Common Chord
Circles \(\omega_1\) and \(\omega_2\) intersect at \(A,B\), and circles \(\omega_2\) and \(\omega_3\) intersect at \(C,D\). Lines \(AB\) and \(CD\) meet at point \(R\). If circles \(\omega_1\) and \(\omega_3\) intersect at \(E,F\), prove that \(R,E,F\) are collinear.
Lines \(AB\), \(CD\), \(EF\) are the radical axes of the corresponding pairs of circles.
Line \(AB\) is the radical axis of \(\omega_1\) and \(\omega_2\), and \(CD\) is the radical axis of \(\omega_2\) and \(\omega_3\). Their intersection \(R\) is the radical center of the three circles. Therefore \(R\) lies on the radical axis of \(\omega_1\) and \(\omega_3\). Since these circles intersect at \(E,F\), their radical axis is line \(EF\). Hence \(R,E,F\) are collinear.
A strong but very clean problem on the radical center.