Problem
GEO-B2-M03-P021 Products Through the Radical Center
#21
★★★★★ Level 5 of 5
Three circles have radical center \(R\). Arbitrary secants through \(R\) are drawn to the circles: they meet \(\omega_1\) at \(A_1,B_1\), \(\omega_2\) at \(A_2,B_2\), and \(\omega_3\) at \(A_3,B_3\). Prove that \(RA_1\cdot RB_1=RA_2\cdot RB_2=RA_3\cdot RB_3\).
At the radical center, the powers with respect to all three circles are equal.
Since \(R\) is the radical center, its powers with respect to \(\omega_1,\omega_2,\omega_3\) are equal. For each circle, the power of point \(R\) equals the product of the segments of any secant through \(R\): \(RA_i\cdot RB_i\). Therefore all three products are equal.
A strong conceptual problem: the radical center does not depend on chosen secants.