Problem
GEO-B2-M03-P019 Radical Center From Products
Through point \(R\), secants are drawn to three circles \(\omega_1,\omega_2,\omega_3\). They give products \(RA_1\cdot RB_1\), \(RA_2\cdot RB_2\), \(RA_3\cdot RB_3\). If these three products are equal, prove that all three radical axes of the pairwise pairs of circles pass through \(R\).
Each product is the power of point \(R\) with respect to the corresponding circle.
The product \(RA_i\cdot RB_i\) equals the power of point \(R\) with respect to circle \(\omega_i\). By the condition, all three powers are equal. Hence \(R\) has equal powers with respect to \(\omega_1\) and \(\omega_2\), with respect to \(\omega_2\) and \(\omega_3\), and with respect to \(\omega_1\) and \(\omega_3\). Therefore \(R\) lies on all three radical axes.
A strong algebraic form of the radical center.