Casey's theorem

Let OO be a circle. Let O1,O2,O3,O4O_{1}, O_{2}, O_{3}, O_{4} be (in that order) four non-intersecting circles that lie inside OO and tangent to it. Denote by tijt_{ij} the length of the exterior common bitangent of the circles Oi,OjO_{i}, O_{j}.
Then: t12.t34+t23.t41=t13.t24t_{12} . t_{34} + t_{23} . t_{41} = t_{13} . t_{24}.

In the degenerate case, where all four circles reduce to points, this is Ptolemy's theorem.