Problem
GEO-B2-M05-P022 A Tangent and a Circle Through the Centre
#22
★★★★★ Level 5 of 5
A circle \(\omega\) passes through \(O\). Line \(l\) is tangent to \(\omega\) at \(T\ne O\) and does not pass through \(O\). Prove that the image of \(l\) is tangent to the image of \(\omega\) at \(T'\).
Tangency is preserved because inversion preserves angles.
At \(T\), the angle between \(l\) and \(\omega\) is \(0^\circ\). Since \(T\ne O\), inversion preserves this angle. Therefore the angle between the images at \(T'\) is also \(0^\circ\), so the image of \(l\) is tangent to the image of \(\omega\).
The problem mixes two basic images: a circle through the centre and a line not through the centre.