Problem
GEO-B2-M03-P024 An Orthogonal Circle and the Radical Center
#24
★★★★★ Level 5 of 5
Circle \(\gamma\) with centre \(X\) is orthogonal to three circles \(\omega_1,\omega_2,\omega_3\). Prove that \(X\) is the radical center of these three circles.
The power of centre \(X\) with respect to each of the three circles equals the square of the radius of \(\gamma\).
Let the radius of \(\gamma\) be \(\rho\). If \(\gamma\) is orthogonal to \(\omega_i(O_i,r_i)\), then \(XO_i^2= ho^2+r_i^2\), hence \(XO_i^2-r_i^2= ho^2\). Thus the powers of point \(X\) with respect to all three circles are equal. Therefore \(X\) lies on every pairwise radical axis, so it is the radical center.
The final problem connects radical axis, radical center, and orthogonal circles.