Large triangles contained in the unit disk of a Minkowski plane

We prove that the unit disk C of an arbitrary Minkowski plane contains an equilateral triangle in at least one of the orientations, whose oriented side lengths are ${frac{3}{2}}$ . We also prove that C permits to inscribe a triangle whose sides are of lengths at least ${frac{3}{2}}$ in the positive orientation, or that they are of lengths at least ${frac{3}{2}}$ in the negative orientation. The ratio ${frac{3}{2}}$ in both the theorems is best possible.