Problem
GEO-B2-M05-P012 Tangent to the Circle of Inversion
#12
★★★☆☆ Level 3 of 5
Line \(l\) is tangent to the circle of inversion at \(T\). Prove that the image of \(l\) is the circle with diameter \(OT\).
The foot from \(O\) to \(l\) is \(T\), and \(T'=T\).
Since \(l\) is tangent to the circle of inversion at \(T\), we have \(OT\perp l\) and \(OT=R\). Thus \(T\) is fixed. By the line-image lemma, the image of \(l\) is the circle with diameter \(OT'\), that is, with diameter \(OT\).
A memorable special case.