A characteristic property of the Euclidean disc

In this paper the following is proved: Let K ⊂ $ \mathbb{E}^2 $ \mathbb{E}^2 be a smooth strictly convex body, and let L ⊂ $ \mathbb{E}^2 $ \mathbb{E}^2 be a line. Assume that for every point x ∈ L/K the two tangent segments from x to K have the same length, and the line joining the two contact points passes through a fixed point in the plane. Then K is an Euclidean disc.