Theorem of the constant chord

constant chord length: ${\displaystyle |P_{1}Q_{1}|=|P_{2}Q_{2}|}$

constant diameter: ${\displaystyle |P_{1}Q_{1}|=|P_{2}Q_{2}|}$

The theorem of the constant chord is a statement of elementary geometry that describes a property of a certain kind of chord of two intersecting circles.

The circles and intersect at the points and and is any point different from and on . The straight lines and intersect the circle in and . The theorem of the constant chord says that the length of the chord of the circle does not depend on the choice of, i.e. is constant. ${\displaystyle k_{1}}$ ${\displaystyle k_{2}}$ ${\displaystyle P}$ ${\displaystyle Q}$ ${\displaystyle Z_{1}}$ ${\displaystyle P}$ ${\displaystyle Q}$ ${\displaystyle k_{1}}$ ${\displaystyle Z_{1}P}$ ${\displaystyle Z_{1}Q}$ ${\displaystyle k_{2}}$ ${\displaystyle P_{1}}$ ${\displaystyle Q_{1}}$ ${\displaystyle P_{1}Q_{1}}$ ${\displaystyle k_{2}}$ ${\displaystyle Z_{1}}$

The sentence remains valid even if it agrees with or , insofar as the undefined straight line or the tangent to in is replaced. ${\displaystyle Z_{1}}$ ${\displaystyle P}$ ${\displaystyle Q}$ ${\displaystyle Z_{1}P}$ ${\displaystyle Z_{1}Q}$ ${\displaystyle k_{1}}$ ${\displaystyle Z_{1}}$

An analogous theorem in three dimensions also applies to the intersection of two spheres. The spheres and have the intersection circle and are any point on the surface of the sphere that does not lie on the intersection circle . The extension of the oblique cone formed by and intersects the sphere in a circle whose diameter has a constant length, i.e. the length of the diameter does not depend on . ${\displaystyle k_{1}}$ ${\displaystyle k_{2}}$ ${\displaystyle k_{s}}$ ${\displaystyle Z_{1}}$ ${\displaystyle k_{1}}$ ${\displaystyle k_{s}}$ ${\displaystyle k_{s}}$ ${\displaystyle Z_{1}}$ ${\displaystyle k_{2}}$ ${\displaystyle Z_{1}}$

Nathan Altshiller-Court described the theorem of the constant tendon in 1925 in the article sur deux cercles secants for the Belgian mathematics journal Mathesis . Eight years later he published the three-dimensional version under the title On Two Intersecting Spheres in the American Mathematical Monthly . Later the sentence found its way into several textbooks, for example in Ross Honsberger's Mathematical Morsels and Roger Nelsen's Proof Without Words II , it can be found as a task and in With harmonic relationships to conics by Halbeisen, Hungerbühler and Läuchli as a proposition.

literature

• Lorenz Halbeisen, Norbert Hungerbühler, Juan Läuchli: With harmonious proportions to conic sections: pearls of classical geometry . Springer 2016, ISBN 9783662530344 , p. 16 ( excerpt )
• 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, Vol. 40, No. 5, 1933, pp. 265-269 ( JSTOR )
• Nathan Altshiller-Court: sur deux cercles secants . Mathesis, Volume 39, 1925, p. 453

Web links

Commons : Constant chord theorem  - collection of images, videos and audio files

• Statement as a task on cut-the-knot.org (English)