Problem
GEO-B2-M03-P008 Three Common Chords
#8
★★★☆☆ Level 3 of 5
Three circles intersect pairwise. The common chord of the first and second circles meets the common chord of the second and third at point \(R\). Prove that \(R\) lies on the common chord of the first and third circles.
The common chord of two circles is their radical axis.
The first common chord is the radical axis of the first and second circles, and the second is the radical axis of the second and third. Their intersection point \(R\) has equal powers with respect to the first, second, and third circles. Therefore \(R\) lies on the radical axis of the first and third circles. Since they intersect, this axis is their common chord.
A classical olympiad formulation of the radical center.