Problem
GEO-B3-M01-P010 Two Circles Through the Center
Circles \(\omega_1\) and \(\omega_2\) pass through points \(A\) and \(B\). An inversion centered at \(A\) is performed. Prove that the images of the circles are two lines meeting at \(B^*\), and that the angle between these lines equals the angle between \(\omega_1\) and \(\omega_2\).
Hint 1. A circle through the inversion center maps to a line.
Hint 2. Point \(B\) belongs to both circles.
E. Full solution. Both circles pass through the inversion center \(A\), so each maps to a line. Since \(B\) lies on both circles, its image \(B^*\) must lie on both image lines; hence the lines meet at \(B^*\). Inversion preserves the angle between circles, so it equals the angle between their images, that is, between the two lines.
This problem prepares the student to turn complicated pencils of circles into pencils of lines.