Some Characteristic Properties of the Circle

Extract

There is a considerable literature on characteristic properties of the circle; a bibliography is listed in Bonnesen-Fenchel. It is proposed here to present some simple characteristic properties of types which are, in a sense, related to one another, as will be seen in what follows.

Principal Definitions and Theorems. The discussion will be restricted to smooth, convex, closed curves. Relative to such a curve, a point, P, will be called

(i) a π-point, if the product of the segments of all chords through P is constant;

(ii) an α-point, if, for every chord through P, the angles, on the same side of the chord, between the chord and the tangents to the curve at the extremities of the chord are equal ; and

(iii) a β-point, if the perpendicular bisectors of all chords through P are concurrent.