Problem
GEO-B2-M05-P020 Two Lines into Two Circles
#20
★★★★☆ Level 4 of 5
Lines \(l\) and \(m\) do not pass through \(O\) and meet at \(P\). Prove that their images are two circles passing through \(O\) and \(P'\), and that the angle between them at \(P'\) equals the angle between \(l\) and \(m\).
Each such line maps to a circle through \(O\); point \(P\) lies on both lines.
The images of \(l\) and \(m\) are circles through \(O\). Since \(P\in l\cap m\), point \(P'\) lies on both image circles. Inversion preserves the angle at \(P\ne O\), so the angle between the circles at \(P'\) equals the angle between lines \(l\) and \(m\).
The reverse picture to two circles through the centre.