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. Formulation of the theorem. Let formula_1 be a circle of radius formula_2. Let formula_3 be (in that order) four non-intersecting circles that lie inside formula_1 and tangent to it. Denote by formula_5 the length of the exterior common bitangent of the circles formula_6. Then: Note that in the degenerate case, where all four circles reduce to points, this is exactly Ptolemy's theorem. Proof. The following proof is attributable to  Zacharias. Denote the radius of circle formula_8 by formula_9 and its tangency point with the circle formula_1 by formula_11. We will use the notation formula_12 for the centers of the circles. Note that from Pythagorean theorem, We will try to express this length in terms of the points formula_14. By the law of cosines in triangle formula_15, Since the circles formula_17 tangent to each other: Let formula_19 be a point on the circle formula_1. According to the law of sines in triangle formula_21: Therefore, and substituting these in the formula above: And finally, the length we seek is We can  now evaluate the left hand side, with the help of the original Ptolemy's theorem applied to the inscribed quadrilateral formula_28: Further generalizations. It can be seen that the four circles need not lie inside the big circle. In fact, they may be tangent to it from the outside as well. In that case, the following change should be made: If formula_6 are both tangent from the same side of formula_1 (both in or both out), formula_5 is the length of the exterior common tangent. If formula_6 are tangent from different sides of formula_1 (one in and one out), formula_5  is the length of the interior common tangent. The converse of Casey's theorem is also true. That is, if equality holds, the circles are tangent to a common circle. Applications. Casey's theorem and its converse can be used to prove a variety of statements in Euclidean geometry. For example, the shortest known proof of Feuerbach's theorem uses the converse theorem.