Problem
GEO-B2-M03-P022 Converse Problem About an Orthogonal Circle
#22
★★★★★ Level 5 of 5
Point \(X\) lies on the radical axis of circles \(\omega_1(O_1,r_1)\) and \(\omega_2(O_2,r_2)\). The common power of point \(X\) with respect to these circles is positive and equals \(\rho^2\). Prove that the circle with centre \(X\) and radius \(\rho\) is orthogonal to both given circles.
From \(XO_i^2-r_i^2= ho^2\), get \(XO_i^2= ho^2+r_i^2\).
Since the common power equals \(\rho^2\), we have \(XO_1^2-r_1^2= ho^2\) and \(XO_2^2-r_2^2= ho^2\). Hence \(XO_1^2= ho^2+r_1^2\) and \(XO_2^2= ho^2+r_2^2\). This is exactly the condition that the circle with centre \(X\) and radius \(\rho\) is orthogonal to each of the given circles.
This problem tests the converse understanding of the relation between the radical axis and orthogonal circles.