Problem
GEO-B2-M03-P001 Common Chord as Radical Axis
#1
★★☆☆☆ Level 2 of 5
Circles \(\omega_1\) and \(\omega_2\) intersect at points \(A\) and \(B\). Prove that their radical axis is line \(AB\).
Take a point \(X\) on \(AB\) and write its power as \(XA\cdot XB\).
For any point \(X\) on line \(AB\), secant \(XAB\) meets both circles at the same points \(A\) and \(B\). Therefore the power of \(X\) with respect to each circle equals \(XA\cdot XB\). Hence the powers are equal, and \(AB\) is the radical axis.
A basic problem: the common chord should be recognised immediately as the radical axis.