Weber problem that, at the optimum, the angles between the AD, BD, and CD straight lines must be equal to 360° / 3 = 120°. Torricelli deduced from that conclusion that: 1– if any triangle ABD, whose ∠ADB angle is equal to 120°, generates an ABDE convex quadrilateral inscribed in a circle, the ∠ABE angle of the ABE triangle must be equal to (180° − 120°)= 60°; 2– one way to determine the set of locations of D for which the ∠ADB angle is equal to 120° is to draw an equilateral ABE triangle (because each angle of an equilateral triangle is
Equilateral dimension equilateral dimension is the maximum number of disjoint translates that can all touch each other. For a normed vector space of dimension "d", the equilateral dimension is at most 2; that is, the L norm has the highest equilateral dimension among all normed spaces. asked whether every normed vector space of dimension "d" has equilateral dimension at least , but this remains unknown. There exist normed spaces in any dimension for which certain sets of four equilateral points cannot be extended to any larger equilateral set but these spaces may have larger equilateral sets that do not include these four