Problem
GEO-B3-M01-P008 Angle Between Circles
Circles \(\omega_1\) and \(\omega_2\) meet at point \(P\), not equal to the inversion center. Prove that the angle between their images at \(P^*\) equals the angle between \(\omega_1\) and \(\omega_2\) at \(P\).
Hint 1. Replace the circles by their tangents at \(P\).
Hint 2. A tangent line maps to a curve whose tangent has the corresponding angle.
E. Full solution. The angle between circles is defined as the angle between their tangents at the intersection point. Let \(l_1\) and \(l_2\) be the tangents to \(\omega_1\) and \(\omega_2\) at \(P\). Inversion sends infinitesimal directions near \(P\) to infinitesimal directions near \(P^*\) and preserves the oriented angle between them; this follows from the similarity of inverse triangles in the limiting position. Hence the angle between the tangents to the images equals the angle between \(l_1\) and \(l_2\).
This is a theoretical problem: it justifies all later angle transfers through inversion.