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.

Proof The following proof is attributable[2] to Zacharias.[3] Denote the radius of circle {estilo de exibição ,O_{eu}} por {estilo de exibição ,R_{eu}} and its tangency point with the circle {estilo de exibição ,O} por {estilo de exibição ,K_{eu}} . We will use the notation {estilo de exibição ,O,O_{eu}} for the centers of the circles. Note that from Pythagorean theorem, {estilo de exibição ,t_{eu j}^{2}={overline {O_{eu}O_{j}}}^{2}-(R_{eu}-R_{j})^{2}.} We will try to express this length in terms of the points {estilo de exibição ,K_{eu},K_{j}} . By the law of cosines in triangle {estilo de exibição ,O_{eu}OO_{j}} , {estilo de exibição {overline {O_{eu}O_{j}}}^{2}={overline {OO_{eu}}}^{2}+{overline {OO_{j}}}^{2}-2{overline {OO_{eu}}}cdot {overline {OO_{j}}}cdot cos angle O_{eu}OO_{j}} Since the circles {estilo de exibição ,O,O_{eu}} tangent to each other: {estilo de exibição {overline {OO_{eu}}}=R-R_{eu},,angle O_{eu}OO_{j}=angle K_{eu}OK_{j}} Deixar {estilo de exibição ,C} be a point on the circle {estilo de exibição ,O} . According to the law of sines in triangle {estilo de exibição ,K_{eu}CK_{j}} : {estilo de exibição {overline {K_{eu}K_{j}}}=2Rcdot sin angle K_{eu}CK_{j}=2Rcdot sin {fratura {angle K_{eu}OK_{j}}{2}}} Portanto, {displaystyle cos angle K_{eu}OK_{j}=1-2sin ^{2}{fratura {angle K_{eu}OK_{j}}{2}}=1-2cdot left({fratura {overline {K_{eu}K_{j}}}{2R}}certo)^{2}=1-{fratura {{overline {K_{eu}K_{j}}}^{2}}{2R^{2}}}} and substituting these in the formula above: {estilo de exibição {overline {O_{eu}O_{j}}}^{2}=(R-R_{eu})^{2}+(R-R_{j})^{2}-2(R-R_{eu})(R-R_{j})deixei(1-{fratura {{overline {K_{eu}K_{j}}}^{2}}{2R^{2}}}certo)} {estilo de exibição {overline {O_{eu}O_{j}}}^{2}=(R-R_{eu})^{2}+(R-R_{j})^{2}-2(R-R_{eu})(R-R_{j})+(R-R_{eu})(R-R_{j})cdot {fratura {{overline {K_{eu}K_{j}}}^{2}}{R^{2}}}} {estilo de exibição {overline {O_{eu}O_{j}}}^{2}=((R-R_{eu})-(R-R_{j}))^{2}+(R-R_{eu})(R-R_{j})cdot {fratura {{overline {K_{eu}K_{j}}}^{2}}{R^{2}}}} And finally, the length we seek is {estilo de exibição t_{eu j}={quadrado {{overline {O_{eu}O_{j}}}^{2}-(R_{eu}-R_{j})^{2}}}={fratura {{quadrado {R-R_{eu}}}cdot {quadrado {R-R_{j}}}cdot {overline {K_{eu}K_{j}}}}{R}}} We can now evaluate the left hand side, with the help of the original Ptolemy's theorem applied to the inscribed quadrilateral {estilo de exibição ,K_{1}K_{2}K_{3}K_{4}} : {estilo de exibição {começar{alinhado}&t_{12}t_{34}+t_{14}t_{23}\[4pt]={}&{fratura {1}{R^{2}}}cdot {quadrado {R-R_{1}}}{quadrado {R-R_{2}}}{quadrado {R-R_{3}}}{quadrado {R-R_{4}}}deixei({overline {K_{1}K_{2}}}cdot {overline {K_{3}K_{4}}}+{overline {K_{1}K_{4}}}cdot {overline {K_{2}K_{3}}}certo)\[4pt]={}&{fratura {1}{R^{2}}}cdot {quadrado {R-R_{1}}}{quadrado {R-R_{2}}}{quadrado {R-R_{3}}}{quadrado {R-R_{4}}}deixei({overline {K_{1}K_{3}}}cdot {overline {K_{2}K_{4}}}certo)\[4pt]={}&t_{13}t_{24}fim{alinhado}}} Further generalizations It can be seen that the four circles need not lie inside the big circle. Na verdade, they may be tangent to it from the outside as well. Nesse caso, the following change should be made:[4] Se {estilo de exibição ,O_{eu},O_{j}} are both tangent from the same side of {estilo de exibição ,O} (both in or both out), {estilo de exibição ,t_{eu j}} is the length of the exterior common tangent.

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