# Konstanter Akkordsatz

Constant chord theorem constant chord length: {Anzeigestil |P_{1}Q_{1}|=|P_{2}Q_{2}|} constant diameter length: {Anzeigestil |P_{1}Q_{1}|=|P_{2}Q_{2}|} The constant chord theorem is a statement in elementary geometry about a property of certain chords in two intersecting circles.

The circles {Anzeigestil k_{1}} und {Anzeigestil k_{2}} intersect in the points {Anzeigestil P} und {Anzeigestil Q} . {Anzeigestil Z_{1}} is an arbitrary point on {Anzeigestil k_{1}} being different from {Anzeigestil P} und {Anzeigestil Q} . The lines {Anzeigestil Z_{1}P} und {Anzeigestil Z_{1}Q} intersect the circle {Anzeigestil k_{2}} in {Anzeigestil P_{1}} und {displaystyle Q_{1}} . The constant chord theorem then states that the length of the chord {Anzeigestil P_{1}Q_{1}} in {Anzeigestil k_{2}} does not depend on the location of {Anzeigestil Z_{1}} an {Anzeigestil k_{1}} , in other words the length is constant.

The theorem stays valid when {Anzeigestil Z_{1}} coincides with {Anzeigestil P} oder {Anzeigestil Q} , provided one replaces the then undefined line {Anzeigestil Z_{1}P} oder {Anzeigestil Z_{1}Q} by the tangent on {Anzeigestil k_{1}} bei {Anzeigestil Z_{1}} .

A similar theorem exists in three dimensions for the intersection of two spheres. The spheres {Anzeigestil k_{1}} und {Anzeigestil k_{2}} intersect in the circle {Anzeigestil k_{s}} . {Anzeigestil Z_{1}} is arbitrary point on the surface of the first sphere {Anzeigestil k_{1}} , that is not on the intersection circle {Anzeigestil k_{s}} . The extended cone created by {Anzeigestil k_{s}} und {Anzeigestil Z_{1}} intersects the second sphere {Anzeigestil k_{2}} in a circle. The length of the diameter of this circle is constant, that is it does not depend on the location of {Anzeigestil Z_{1}} an {Anzeigestil k_{1}} .

Nathan Altshiller Court described the constant chord theorem 1925 in the article sur deux cercles secants for the Belgian math journal Mathesis. Eight years later he published On Two Intersecting Spheres in the American Mathematical Monthly, which contained the 3-dimensional version. Later it was included in several textbooks, such as Ross Honsberger's Mathematical Morsels and Roger B. Nelsen's Proof Without Words II, where it was given as a problem, or the German geometry textbook Mit harmonischen Verhältnissen zu Kegelschnitten by Halbeisen, Hungerbühler and Läuchli, where it was given as a theorem.

References Lorenz Halbeisen, Norbert Hungerbühler, Juan Läuchli: Mit harmonischen Verhältnissen zu Kegelschnitten: Perlen der klassischen Geometrie. Springer 2016, ISBN 9783662530344, p. 16 (Deutsch) Roger B. Nelsen: Proof Without Words II. MAA, 2000, p. 29 Ross Honsberger: Mathematical Morsels. MAA, 1979, ISBN 978-0883853030, pp. 126–127 Nathan Altshiller Court: On Two Intersecting Spheres. The American Mathematical Monthly, Band 40, Nr. 5, 1933, pp. 265–269 (JSTOR) Nathan Altshiller-Court: sur deux cercles secants. Mathesis, Band 39, 1925, p. 453 (Französisch) External links Wikimedia Commons has media related to Constant chord theorem. constant chord theorem as problem at cut-the-knot.org Categories: Theorems about circlesEuclidean geometry

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