There are two circles. There are also two common "outside tangents" (they are marked red in the diagram) and one "inside tangent" which is marked blue. (I am not a native English speaker, I don't know how to call the outside and the inside tangents correctly) The inside tangent intersects the circles in points A and B and intersects two outside tangents in points A1 and B1. How can I prove that the line segment AA1 equals BB1?
Let $E$ be the point of tangency of the lower tangent and the circle on the left. Then $AA_1 = AE$.
Let $F$ be the point of tangency of the lower tangent and the circle on the right.
$AA_1 = EF - A_1B$. $\ BB_1 = E_1F_1 (upper circle) - AB_1$.
Now $AA_1 = A_1B_1 - AB_1$; $BB_1 = A_1B_1 - A_1B$.
From these equations we get that $AB_1 = A_1B$.