Problem
GEO-B2-M05-P019 Tangency at the Centre
#19
★★★★☆ Level 4 of 5
Two circles pass through \(O\) and are tangent to each other at \(O\). Prove that their images under an inversion with centre \(O\) are parallel lines.
Both circles map to lines. Their tangents at \(O\) coincide.
A circle through \(O\) maps to a line perpendicular to the line joining \(O\) with its centre. If two circles are tangent at \(O\), their centres lie on one line perpendicular to the common tangent. Therefore the two image lines are perpendicular to the same line. Hence they are parallel.
A nontrivial but classic image under inversion.