Problem
GEO-B3-M01-P009 Tangency at the Inversion Center
#9
★★☆☆☆ Level 2 of 5
Two circles are tangent at point \(A\). Prove that under any inversion centered at \(A\), they map to two parallel lines.
Inspired by Prasolov inversion method
Hint 1. Both circles pass through the inversion center.
Hint 2. Tangency means a zero angle between the images.
E. Full solution. Since both circles pass through the inversion center \(A\), each maps to a line. The original circles are tangent at \(A\), so the angle between them is \(0^\circ\). Inversion preserves the angle between curves; hence the angle between the resulting lines is also \(0^\circ\). Two distinct lines with zero angle are parallel.
This is one of the most useful moves of the module: tangency at the inversion center becomes parallelism.