Equichordal point

In geometry, an equichordal point is a point defined relative to a convex plane curve such that all chords passing through the point are equal in length. Two common figures with equichordal points are the circle and the limaçon.

In 1916 Fujiwara proposed the question of whether a curve could have two equichordal points (offering in the same paper a proof that three or more is impossible). The problem remained unsolved until it was finally proven impossible in 1996 by Marek Rychlik. Fujiwara's problem is known as the Equichordal Point Problem.

Despite its elementary formulation, the Equichordal Point Problem was difficult to solve. Rychlik's Theorem is proved by methods of advanced complex analysis and algebraic geometry and it is 72 pages long. It is believed unlikely that a short, simple proof exists. It would be surprising if such a proof were found.

References

• "Equichordal Point" in Wolfram MathWorld