J. Geom. 95 (2009), 31–39 c  2009 Birkh¨auser Verlag Basel/Switzerland 0047-2468/09/010031-9 published online December 4, 2009 DOI 10.1007/s00022-009-0019-1 Journal of Geometry Large triangles contained in the unit disk of a Minkowski plane Ewa Fabi´ nska and Marek Lassak Abstract. 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 3 2 . We also prove that C permits to inscribe a triangle whose sides are of lengths at least 3 2 in the positive orientation, or that they are of lengths at least 3 2 in the negative orientation. The ratio 3 2 in both the theorems is best possible. Mathematics Subject Classification (2000). 52A10, 52A21. Keywords. Minkowski plane, unit disk, triangle. We start by recalling the notion of the Minkowski plane. Let C be a convex body in Euclidean plane E 2 and let z ∈ int(C). We define the oriented dis- tance δ C,z (a, b) of points a and b as δ C,z (a, b)= |ab|/|zw|, where w ∈ bd(C) such that the vectors −→ zw and −→ ab have the same orientation (the symbol || stands for the Euclidean distance). We call C the unit disk and z the origin. The plane with the oriented distance is called Minkowski plane. It is a special case of Minkowski space. Many properties of Minkowski plane are presented in [5, 6, 9]. When C is centrally symmetric with z as its center, we get the two-dimensional normed plane. The well-known problem if the self-circumference of an arbitrary Minkow- ski unit disk is at least 6 (see [2, 5, 7]) caused many questions about “large” inscribed polygons in the unit disk. Our particular interest is in “large” inscribed and also contained triangles. We continue our research from [4]. If we go on the boundary of a triangle according to the positive (respectively: negative) orientation and if the oriented lengths of its sides are equal, then we call the triangle equilateral in the positive (respectively: negative) orienta- tion. We also simply say orientation when it is positive. Since the Minkowski functional is non-symmetric in general, equilateral triangles usually are not