Problem
GEO-B2-M05-P018 Circle Through a Chosen Point
#18
★★★★☆ Level 4 of 5
Points \(A\) and \(B\) lie on a ray from \(O\). An inversion with centre \(O\) is chosen so that \(A\) maps to \(B\). A circle \(\omega\) passes through \(O\) and \(A\). Prove that the image of \(\omega\) is a line passing through \(B\).
A circle through \(O\) maps to a line, and \(A\) maps to \(B\).
Since \(\omega\) passes through the centre of inversion, its image is a line. Point \(A\) lies on \(\omega\), and its image by the condition is \(B\). Therefore the image line passes through \(B\).
Shows the purpose of choosing the radius for a pair of points.