Problem
GEO-B2-M10-P020 Homothety Centre of Two Circles
Two disjoint circles of different radii are given. Their common external tangents touch the first circle at \(A,B\) and the second at \(C,D\), with \(A,C\) on one tangent and \(B,D\) on the other. Prove that lines \(AC\) and \(BD\) meet on the line of the centres of the circles.
Recall the external homothety centre of two circles.
There is an external homothety sending the first circle to the second; its centre \(X\) lies on the line of the centres. A common external tangent maps to itself, so the tangency point \(A\) maps to \(C\), and \(B\) maps to \(D\). Therefore \(A,X,C\) are collinear and \(B,X,D\) are collinear. Hence lines \(AC\) and \(BD\) meet at \(X\) on the line of the centres.
The problem reinforces the geometric meaning of a homothety centre.