Constant chord theorem

The constant chord theorem is a statement in elementary geometry about a property of certain chords in two intersecting circles.

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

The theorem stays valid when ${\displaystyle Z_{1}}$ coincides with ${\displaystyle P}$ or ${\displaystyle Q}$, provided one replaces the then undefined line ${\displaystyle Z_{1}P}$ or ${\displaystyle Z_{1}Q}$ by the tangent on ${\displaystyle k_{1}}$ at ${\displaystyle Z_{1}}$.

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

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.

• Lorenz Halbeisen, Norbert Hungerbühler, Juan Läuchli: Mit harmonischen Verhältnissen zu Kegelschnitten: Perlen der klassischen Geometrie. Springer 2016, ISBN 9783662530344, p. 16 (German)
• 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 (French)

• constant chord theorem as problem at cut-the-knot.org