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.

be a circle of radius

be (in that order) four non-intersecting circles that lie inside

the length of the exterior common bitangent of the circles

Then:[1] Note that in the degenerate case, where all four circles reduce to points, this is exactly Ptolemy's theorem.

The following proof is attributable[2] to Zacharias.

[3] Denote the radius of circle

and its tangency point with the circle

for the centers of the circles.

Note that from Pythagorean theorem, We will try to express this length in terms of the points

By the law of cosines in triangle

According to the law of sines in triangle

: 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

: It can be seen that the four circles need not lie inside the big circle.

In that case, the following change should be made:[4] If

is the length of the exterior common tangent.

is the length of the interior common tangent.

The converse of Casey's theorem is also true.

[4] That is, if equality holds, the circles are tangent to a common circle.

Casey's theorem and its converse can be used to prove a variety of statements in Euclidean geometry.

For example, the shortest known proof[1]: 411  of Feuerbach's theorem uses the converse theorem.