Problem
GEO-B2-M03-P015 Centre of an Orthogonal Circle
#15
★★★★☆ Level 4 of 5
Circle \(\gamma\) with centre \(X\) and radius \(\rho\) is orthogonal to two circles \(\omega_1(O_1,r_1)\) and \(\omega_2(O_2,r_2)\). Prove that \(X\) lies on the radical axis of \(\omega_1\) and \(\omega_2\).
For orthogonal circles, \(XO_i^2= ho^2+r_i^2\).
From orthogonality of \(\gamma\) and \(\omega_1\), we get \(XO_1^2= ho^2+r_1^2\), hence \(XO_1^2-r_1^2= ho^2\). Similarly, \(XO_2^2-r_2^2= ho^2\). Therefore the powers of point \(X\) with respect to \(\omega_1\) and \(\omega_2\) are equal, and \(X\) lies on the radical axis.
This is already a stronger idea: the radical axis appears through orthogonality of circles.