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We show that, in any ordered plane with a symmetric orthogonality relation which allows for a meaningful definition of acute and obtuse angles, in which all points are colored with three colors, such that each color is used at least once, there must exist both an acute triangle whose vertices have all three colors and an obtuse triangle with the same property. We also show that, in both a geometry endowed with an orthogonality relation, in which there is a reflection in every line, in which all right angles are bisectable, which satisfies Bachmann’s Lotschnittaxiom (the perpendiculars raised on the sides of a right angle intersect), and in plane absolute geometry, in which all points are colored with three colors, such that each color is used at least once, there exists a right triangle with all vertices of different colors.

A set X in a real Hilbert space H is called an a t-set if every three-element subset of X forms either an acute-angled triangle or a right-angled triangle. The maximal cardinality of an a t-set in an infinite-dimensional H is found. Furthermore, the number of right angles in the unit cube [ 0 , 1 ] n ${[0,1]^n}$ is calculated. As an application, a simple solution of a well-known problem is given, concerning the maximal cardinality of a strong a t-set in the Euclidean space ℝ n .