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Equichordal Point


EquichordalCurves

An equichordal point is a point p for which all the chords of a curve C passing through p are of the same length. In other words, p is an equichordal point if, for every chord [x,y] of length p of the curve C, p satisfies

 |x-p|+|y-p|=p.
(1)

A function r(theta) satisfying

 r(0)=p-r(pi)
(2)

corresponds to a curve with equichordal point (0, 0) and chord length p defined by letting r(theta) be the polar equation of the half-curve for 0<=theta<=pi and then superimposing the polar equation r(theta)-p over the same range. The curves illustrated above correspond to polar equations of the form

 r(theta)=x+(1/2-x)cos(2theta)
(3)

for various values of x.

Although it long remained an outstanding problem (the equichordal point problem), it is now known that a planar convex region can not have two equichordal points (Rychlik 1997).


See also

Chord, Equichordal Point Problem, Equiproduct Point, Equireciprocal Point

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References

Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, p. 9, 1991.Dirac, G. A. "Ovals with Equichordal Points." J. London Math. Soc. 27, 429-437, 1952.Hallstrom, A. P. "Equichordal and Equireciprocal Points." Bogasici Univ. J. Sci. 2, 83-88, 1974.Rychlik, M. "The Equichordal Point Problem." Elec. Res. Announcements Amer. Math. Soc. 2, 108-123, 1996.Rychlik, M. "A Complete Solution to the Equichordal Problem of Fujiwara, Blaschke, Rothe, and Weitzenböck." Invent. Math. 129, 141-212, 1997.Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, p. 152, 1999.Zindler, K. "Über konvexe Gebilde, II." Monatshefte f. Math. u. Phys. 3, 25-29, 1921.

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Equichordal Point

Cite this as:

Weisstein, Eric W. "Equichordal Point." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EquichordalPoint.html

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