Problem
GEO-B2-M04-P003 Tangent Circles
#3
★★☆☆☆ Level 2 of 5
Two circles are externally tangent at \(T\). Their centres are \(O_1\) and \(O_2\), and their radii are \(3\) and \(7\). Prove that \(T\) is the centre of a homothety sending one circle to the other, and find \(TO_1:TO_2\).
The point of tangency lies on the line of centres, and the distances to the centres are the radii.
For external tangency, points \(O_1,T,O_2\) are collinear, with \(TO_1=3\) and \(TO_2=7\). The homothety with centre \(T\) and ratio \(-\frac{7}{3}\) sends centre \(O_1\) to \(O_2\) and the circle of radius \(3\) to the circle of radius \(7\). Therefore \(T\) is the centre of such a homothety, and \(TO_1:TO_2=3:7\).
It is useful to discuss the sign of the homothety ratio in external tangency.