Casey's theorem

In mathematics, Casey's theorem, also known as the generalized Ptolemy's theorem, is a theorem in Euclidean geometry named after the Irish mathematician John Casey.

Let ${\displaystyle \,O}$ be a circle of radius ${\displaystyle \,R}$. Let ${\displaystyle \,O_{1},O_{2},O_{3},O_{4}}$ be (in that order) four non-intersecting circles that lie inside ${\displaystyle \,O}$ and tangent to it. Denote by ${\displaystyle \,t_{ij}}$ the length of the exterior common bitangent of the circles ${\displaystyle \,O_{i},O_{j}}$. Then:

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