Théorème d'accord constant

Constant chord theorem constant chord length: {style d'affichage |P_{1}Q_{1}|=|P_{2}Q_{2}|} constant diameter length: {style d'affichage |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 {style d'affichage k_{1}} et {style d'affichage k_{2}} intersect in the points {style d'affichage P} et {style d'affichage Q} . {style d'affichage Z_{1}} is an arbitrary point on {style d'affichage k_{1}} being different from {style d'affichage P} et {style d'affichage Q} . Les lignes {style d'affichage Z_{1}P} et {style d'affichage Z_{1}Q} intersect the circle {style d'affichage k_{2}} dans {style d'affichage P_{1}} et {style d'affichage Q_{1}} . The constant chord theorem then states that the length of the chord {style d'affichage P_{1}Q_{1}} dans {style d'affichage k_{2}} does not depend on the location of {style d'affichage Z_{1}} sur {style d'affichage k_{1}} , in other words the length is constant.

The theorem stays valid when {style d'affichage Z_{1}} coincides with {style d'affichage P} ou {style d'affichage Q} , provided one replaces the then undefined line {style d'affichage Z_{1}P} ou {style d'affichage Z_{1}Q} by the tangent on {style d'affichage k_{1}} à {style d'affichage Z_{1}} .

A similar theorem exists in three dimensions for the intersection of two spheres. The spheres {style d'affichage k_{1}} et {style d'affichage k_{2}} intersect in the circle {style d'affichage k_{s}} . {style d'affichage Z_{1}} is arbitrary point on the surface of the first sphere {style d'affichage k_{1}} , that is not on the intersection circle {style d'affichage k_{s}} . The extended cone created by {style d'affichage k_{s}} et {style d'affichage Z_{1}} intersects the second sphere {style d'affichage 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 {style d'affichage Z_{1}} sur {style d'affichage 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.

Références Lorenz Halbeisen, Norbert Hungerbühler, Juan Lauchli: Avec des relations harmoniques aux sections coniques: Perles de géométrie classique. Springer 2016, ISBN 9783662530344, p. 16 (Allemand) Roger B.. Nelson: Proof Without Words II. MAA, 2000, p. 29 Ross Honsberg: Mathematical Morsels. MAA, 1979, ISBN 978-0883853030, pp. 126–127 Nathan Altshiller Court: On Two Intersecting Spheres. Le mensuel mathématique américain, Band 40, Nr. 5, 1933, pp. 265–269 (JSTOR) Nathan Altshiller-Court: sur deux cercles secants. Mathesis, Band 39, 1925, p. 453 (Français) 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