KnowraPtolemy's theoremLinked fromLinked fromThe 16 pages that link to Ptolemy's theorem, each with the reason it gives.All 16Related 9Compared with 7Law of cosinesCompared with: It is a distinct metric theorem sometimes used to derive triangle side relations.Cyclic quadrilateralRelated: It links all four side lengths to both diagonals when the vertices are concyclic.Brahmagupta's formulaRelated: Its relation among sides and diagonals helps derive the area formula.Ptolemy's inequalityCompared with: This is the equality case of the inequality, characterizing when its bound is sharp.Bretschneider's formulaRelated: It offers another route from cyclic geometry to Brahmagupta's area result.Tangent–secant theoremRelated: Both results turn cyclic geometry into length equations, though Ptolemy’s theorem concerns quadrilaterals.Stewart's theoremRelated: Like Stewart’s theorem, it converts a geometric configuration into an exact length identity.Euler's quadrilateral theoremCompared with: It gives a different diagonal-side relation, but only for cyclic quadrilaterals.Inversive geometryRelated: Inversion can establish this relation by sending a vertex to infinity and simplifying the cyclic figure.Pedoe's inequalityCompared with: It is another sharp geometric relation, but concerns cyclic quadrilaterals rather than paired triangles.Clifford's circle theoremsCompared with: It concerns metric lengths on a circle, rather than incidence across successive circles.Casey's theoremRelated: Casey's relation replaces vertex distances with common-tangent lengths.Five circles theoremCompared with: It concerns the same circle geometry but gives a metric identity instead of a new concyclicity result.Haruki's theoremRelated: Cyclic quadrilaterals can turn circle intersections into tractable segment equations.Pompeiu's theoremCompared with: Both relate distances in cyclic configurations, but Ptolemy's theorem requires a quadrilateral.Van Schooten's theoremRelated: Both results convert cyclic point configurations into exact equations among segment lengths.
KnowraPtolemy's theoremLinked fromLinked fromThe 16 pages that link to Ptolemy's theorem, each with the reason it gives.All 16Related 9Compared with 7Law of cosinesCompared with: It is a distinct metric theorem sometimes used to derive triangle side relations.Cyclic quadrilateralRelated: It links all four side lengths to both diagonals when the vertices are concyclic.Brahmagupta's formulaRelated: Its relation among sides and diagonals helps derive the area formula.Ptolemy's inequalityCompared with: This is the equality case of the inequality, characterizing when its bound is sharp.Bretschneider's formulaRelated: It offers another route from cyclic geometry to Brahmagupta's area result.Tangent–secant theoremRelated: Both results turn cyclic geometry into length equations, though Ptolemy’s theorem concerns quadrilaterals.Stewart's theoremRelated: Like Stewart’s theorem, it converts a geometric configuration into an exact length identity.Euler's quadrilateral theoremCompared with: It gives a different diagonal-side relation, but only for cyclic quadrilaterals.Inversive geometryRelated: Inversion can establish this relation by sending a vertex to infinity and simplifying the cyclic figure.Pedoe's inequalityCompared with: It is another sharp geometric relation, but concerns cyclic quadrilaterals rather than paired triangles.Clifford's circle theoremsCompared with: It concerns metric lengths on a circle, rather than incidence across successive circles.Casey's theoremRelated: Casey's relation replaces vertex distances with common-tangent lengths.Five circles theoremCompared with: It concerns the same circle geometry but gives a metric identity instead of a new concyclicity result.Haruki's theoremRelated: Cyclic quadrilaterals can turn circle intersections into tractable segment equations.Pompeiu's theoremCompared with: Both relate distances in cyclic configurations, but Ptolemy's theorem requires a quadrilateral.Van Schooten's theoremRelated: Both results convert cyclic point configurations into exact equations among segment lengths.