Consider a fixed angle \(\alpha \in (0,\pi )\) and a disc B. All the chords \([z_1,z_2]\) of B such that the tangent lines of B at \(z_1\) and \(z_2\) intersect at an angle \(\alpha \) ( \(\alpha \) -chords), have the same length. It is known that under certain circumstances, a convex body with \(\alpha \) -chords of constant length is a disc. It is also known that a convex body, whose isoptic chords have constant length for angles \(\alpha \) and \(\pi -\alpha \) must be a disc, when \(\frac{\alpha }{\pi }\) is an irrational number or \(\alpha =\frac{p}{2r+1}\pi \) , where p and r are natural numbers. In this article we present a proof for the general case of this last statement. The set of points under which a convex body K is seen at an angle \(\alpha \) is known as the isoptic of angle \(\alpha \) for K. The pair of isoptics of angles \(\alpha \) and \(\pi -\alpha \) is called a bisoptic for K. We also give some characterizations of the disc and figures of constant widht in terms of properties of their bisoptics.
