Problem
GEO-B2-M04-P021 Two Common Tangents
Two disjoint circles have centres \(O_1\) and \(O_2\). Their external common tangents meet at \(H\). Prove that \(H,O_1,O_2\) are collinear.
From \(H\), two tangents are drawn to each circle. Compare the right triangles formed by radii to the tangency points.
Let the external tangents touch the first circle at \(A_1,B_1\) and the second at \(A_2,B_2\), with \(A_1,A_2,H\) on one tangent and \(B_1,B_2,H\) on the other. Radii to points of tangency are perpendicular to the tangents, so triangles \(HO_1A_1\) and \(HO_2A_2\) are right triangles and share the angle at \(H\). Hence they are similar. Thus \(\frac{HO_1}{HO_2}=\frac{O_1A_1}{O_2A_2}\). The same homothety with centre \(H\) sends the radius of the first circle to the corresponding radius of the second. Therefore it sends the first circle to the second and \(O_1\) to \(O_2\). Hence \(H,O_1,O_2\) are collinear.
A strong problem: external common tangents naturally lead to the external centre of homothety.