Casey's theorem

Casey's theorem relates the lengths of common tangents between four circles tangent to a fifth circle. It generalizes Ptolemy's theorem for cyclic quadrilaterals.

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Common tangent: A line tangent to two circles at distinct points. The theorem measures lengths of common tangent segments between pairs of its four circles.
External tangency: Two circles are externally tangent when they touch at one point without either containing the other. External tangencies determine the sign and value of several pairwise tangent lengths.
John Casey: An Irish mathematician known for work on geometry and for the theorem bearing his name. Casey's circle-geometry work supplied the result now called Casey's theorem.
Ptolemy's inequality: For any quadrilateral, the product of its diagonals is at most the sum of the products of opposite sides. It extends the equality principle underlying Casey's theorem beyond cyclic configurations.
Oriented circle: A circle equipped with a choice of orientation, often encoded by a signed radius. Orientations distinguish internal from external tangency in general forms of the theorem.
Internal tangency: Two circles are internally tangent when one lies inside the other and their boundaries touch once. Internal tangencies change the signed lengths used in the generalized relation.
Ptolemy: An ancient Greek mathematician and astronomer whose geometric results include a theorem on cyclic quadrilaterals. His theorem is the classical special case generalized by Casey.
Miquel's theorem: A theorem asserting concyclicity of four circumcenters in a configuration of intersecting circles. It offers a complementary circle-configuration result used alongside tangent-circle geometry.
Tangency: The condition in which geometric objects meet with matching tangent directions at their point of contact. The five-circle configuration is defined by tangency to one common circle.
Circle inversion: A transformation that maps points through a fixed circle while exchanging its interior and exterior. Inversion can normalize a common tangent circle and simplify configurations of tangent circles.