Casey's theorem

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.

Conteúdo 1 Formulation of the theorem 2 Prova 3 Generalizações adicionais 4 Formulários 5 Referências 6 External links Formulation of the theorem {estilo de exibição t_{12}cdot t_{34}+t_{14}cdot t_{23}-t_{13}cdot t_{24}=0} Deixar {estilo de exibição ,O} be a circle of radius {estilo de exibição ,R} . Deixar {estilo de exibição ,O_{1},O_{2},O_{3},O_{4}} be (in that order) four non-intersecting circles that lie inside {estilo de exibição ,O} and tangent to it. Denote by {estilo de exibição ,t_{eu j}} the length of the exterior common bitangent of the circles {estilo de exibição ,O_{eu},O_{j}} . Então:[1] {estilo de exibição ,t_{12}cdot t_{34}+t_{14}cdot t_{23}=t_{13}cdot t_{24}.} Note that in the degenerate case, where all four circles reduce to points, this is exactly Ptolemy's theorem.

Se {estilo de exibição ,O_{eu},O_{j}} are tangent from different sides of {estilo de exibição ,O} (one in and one out), {estilo de exibição ,t_{eu j}} is the length of the interior common tangent.

The converse of Casey's theorem is also true.[4] Aquilo é, 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. Por exemplo, the shortest known proof[1]: 411  of Feuerbach's theorem uses the converse theorem.

Referências ^ Ir para: a b Casey, J. (1866). "On the Equations and Properties: (1) of the System of Circles Touching Three Circles in a Plane; (2) of the System of Spheres Touching Four Spheres in Space; (3) of the System of Circles Touching Three Circles on a Sphere; (4) of the System of Conics Inscribed to a Conic, and Touching Three Inscribed Conics in a Plane". Anais da Academia Real Irlandesa. 9: 396-423. JSTOR 20488927. ^ Bottema, O. (1944). Hoofdstukken uit de Elementaire Meetkunde. (translation by Reinie Erné as Topics in Elementary Geometry, Springer 2008, of the second extended edition published by Epsilon-Uitgaven 1987). ^ Zacharias, M. (1942). "Der Caseysche Satz". Relatório anual da Associação Alemã de Matemáticos. 52: 79-89. ^ Saltar para: a b Johnson, Roger A. (1929). Modern Geometry. Houghton Mifflin, Boston (republished facsimile by Dover 1960, 2007 as Advanced Euclidean Geometry). External links Wikimedia Commons has media related to Casey's theorem. Weisstein, Eric W. "Casey's theorem". MathWorld. Shailesh Shirali: "'On a generalized Ptolemy Theorem'". Dentro: Crux Mathematicorum, Volume. 22, Não. 2, pp. 49-53 Categorias: Theorems about circlesEuclidean geometry

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