Problem
GEO-B2-M03-P007 Radical Center Theorem
#7
★★★☆☆ Level 3 of 5
The radical axes of circles \(\omega_1,\omega_2\) and \(\omega_2,\omega_3\) meet at point \(R\). Prove that \(R\) lies on the radical axis of circles \(\omega_1\) and \(\omega_3\).
Write the equalities of powers that follow from \(R\) lying on two radical axes.
From \(R\in\operatorname{Rad}(\omega_1,\omega_2)\), we get \(\operatorname{Pow}_{\omega_1}(R)=\operatorname{Pow}_{\omega_2}(R)\). From \(R\in\operatorname{Rad}(\omega_2,\omega_3)\), we get \(\operatorname{Pow}_{\omega_2}(R)=\operatorname{Pow}_{\omega_3}(R)\). Therefore \(\operatorname{Pow}_{\omega_1}(R)=\operatorname{Pow}_{\omega_3}(R)\), so \(R\) lies on the third radical axis.
This is the main mechanism in all three-circle problems.