Problem
GEO-B3-M01-P007 Tangency After Inversion
Circles \(\omega_1\) and \(\omega_2\) are tangent at point \(T\), with \(T\neq O\). Prove that their images under inversion centered at \(O\) are also tangent.
Hint 1. Compare the angle between the circles at \(T\).
Hint 2. Inversion preserves the angle between tangents.
E. Full solution. At the tangency point the circles have a common tangent, so the angle between them is \(0^\circ\). Inversion preserves angles between curves, meaning angles between tangents. Point \(T\) maps to \(T^*\), and the tangents to the images at \(T^*\) form the same angle \(0^\circ\). Therefore the images have a common tangent at \(T^*\), so they are tangent.
The problem reinforces an important principle: inversion does not destroy tangency if the tangency point is not the center.