Problem
GEO-B2-M05-P016 An Orthogonal Circle
#16
★★★★☆ Level 4 of 5
A circle \(\omega\) with centre \(C\) and radius \(r\) is orthogonal to the circle of inversion with centre \(O\) and radius \(R\). Prove that \(\omega\) maps to itself.
Draw a secant through \(O\), meeting \(\omega\) at \(A\) and \(B\), and find \(OA\cdot OB\).
Orthogonality gives \(OC^2=R^2+r^2\). For a secant \(OAB\), by the power of point \(O\) with respect to \(\omega\), \(OA\cdot OB=OC^2-r^2=R^2\). Thus \(A\) and \(B\) are inverse points. Every secant through \(O\) swaps points of the circle in pairs, so \(\omega\) maps to itself.
A strong link between inversion and power of a point.