In Euclidean plane geometry, the equichordal point problem is the question whether a closed planar convex body can have two equichordal points. The problem was originally posed in 1916 by Fujiwara and in 1917 by Wilhelm Blaschke, Hermann Rothe, and Roland Weitzenböck. A generalization of this problem statement was answered in the negative in 1997 by Marek R. Rychlik. Problem statement An equichordal curve is a closed planar curve for which a point in the plane exists such that all chords passing through this point are equal in length. Such a point is called an equichordal point. It is easy to construct equichordal curves with a single equichordal point, particularly when the curves are symmetric; the simplest construction is a circle. It has long only been conjectured that no convex equichordal curve with two equichordal points can exist. More generally, it was asked whether there exists a Jordan curve with two equichordal points and , such that the curve would be star-shaped with respect to each of the two points. Excentricity (or eccentricity) Many results on equichordal curves refer to their excentricity. It turns out that the smaller the excentricity, the harder it is to disprove the existence of curves with two equichordal points. It can be shown rigorously that a small excentricity means that the curve must be close to the circle. Let be the hypothetical convex curve with two equichordal points and . Let be the common length of all chords of the curve passing through or . Then excentricity is the ratio where is the distance between the points and . The history of the problem The problem has been extensively studied, with significant papers published over eight decades preceding its solution: In 1916 Fujiwara proved that no convex curves with three equichordal points exist. In 1917 Blaschke, Rothe and Weitzenböck formulated the problem again. In 1923 Süss showed certain symmetries and uniqueness of the curve, if it existed. In 1953 G. A. Dirac showed some explicit bounds on the curve, if it existed. In 1958 Wirsing showed that the curve, if it exists, must be an analytic curve. In this deep paper, he correctly identified the problem as perturbation problem beyond all orders. In 1966 Ehrhart proved that there are no equichordal curves with excentricities > 0.5. In 1988 Michelacci proved that there are no equichordal curves with excentricities > 0.33. The proof is mildly computer-assisted.