Equihord curve and its equichord center.
The equihord center is a point inside a flat curve such that all the chords passing through it are equal. Curves having an equihord center are called equihord curves.
Examples
Equihord curves are
- A circle , the center of a circle is its equichord center.
- Podera on a curve of constant width relative to a point
inside the curve. Wherein
- In particular, for a circle we get a Pascal snail .
Properties
- Any convex curve has at most one equihord center.
- That a convex curve cannot have three centers was proved by Fujiwara in 1916; he formulated the problem that there cannot be two either. The problem was independently formulated by Wilhelm Blaschke , and in 1917 and solved by Marek Rychlik in 1997. His proof is rather complicated, it takes 72 pages and uses complex analysis and algebraic geometry .
Literature
- W. Blaschke, W. Rothe, and R. Weitztenböck. Aufgabe 552. Arch. Math. Phys., 27:82, 1917
- M. Fujiwara. Über die Mittelkurve zweier geschlossenen konvexen Curven in Bezug auf einen Punkt. Tôhoku Math J., 10: 99–103, 1916
- Marek R. Rychlik (1997), "A complete solution to the equichordal point problem of Fujiwara, Blaschke, Rothe and Weitzenböck", Inventiones Mathematicae 129 (1): 141–212
- Steven G. Krantz (1997), Techniques of Problem Solving, American Mathematical Society,