Quadratic ellipses in Hilbert geometries

A Hilbert plane is a model of the hyperbolic plane if and only if at least one circle is a quadric. Further, a Hilbert plane is a model of the hyperbolic plane if and only if at least two confocal ellipses are quadrics, or at least one ellipse with numerical eccentricity greater than \(1/\sqrt{3}\) $1/\sqrt{3}$ is a quadric.