A Maximal Parallelogram Characterization of Ovals Having Circles as Orthoptic Curves

It is shown that ovals admitting circles as orthoptic curves are precisely characterized by the property that every one of their points is the vertex of exactly one maximal-perimeter inscribed parallelogram. This generalizes an old property of ellipses, most recently revived by Connes and Zagier in the paper A Property of Parallelograms Inscribed in Ellipses. Let C be a centrally symmetric smooth strictly convex closed plane curve (oval with a center). It has been known for a long time, see for instance [1], that if C is an ellipse then among all the inscribed parallelograms those of maximal perimeter have the property that any point of C is the vertex of exactly one. The proof given in [1] makes it clear that this property is related to the fact that the Monge orthoptic curve, i.e., the locus of all the points from where a given closed curve can be seen at a right angle, of an ellipse is a circle. The purpose of this note is to show that the maximal-perimeter property of parallelograms inscribed in centrally symmetric ovals described above for ellipses is characteristic precisely to the class of ovals admitting circles as orthoptic curves. For a parallel result, proved by analytic methods, see [2]. Theorem 1. Let C be a centrally symmetric oval. Then every point of C is the vertex of an unique parallelogram of maximal perimeter among those inscribed in C if and only if the orthoptic curve of C is a circle. The proof of Theorem 1 will be an immediate consequence of the following lemma. Lemma 2. Let ABCD be a parallelogram of maximal perimeter among those inscribed in a given centrally symmetric oval C. Then the tangent lines to C at A, B, C, and D, form with the sides of the parallelogram equal angles, respectively, (for the ‘table’ C, the parallelogram is a billiard of period 4), and intersect at the vertices of a rectangle PQRS, concentric to C (see Figure 1). Moreover, the perimeter of the parallelogram ABCD equals four times the radius of the circle circumscribed about the rectangle PQRS. Proof. The strict convexity and central symmetry of C imply that the center of any parallelogram inscribed in C is the same as the center O of C. Therefore, any parallelogram inscribed in C is completely determined by two consecutive vertices. The function from C × C to [0,∞) which gives the perimeter of the associated Publication Date: March 1, 2010. Communicating Editor: Paul Yiu.