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Similar to a Pythagorean triple, an Eisenstein triple (named after Gotthold Eisenstein) is a set of integers which are the lengths of the sides of a triangle where one of the angles is 60 or 120 degrees. The relation of such triangles to the Eisenstein integers is analogous to the relation of Pythagorean triples to the Gaussian integers.
The two isogonic centers are the intersections of three vesicae piscis whose paired vertices are the vertices of the triangle. When the largest angle of the triangle is not larger than 120°, X(13) is the Fermat point. The angles subtended by the sides of the triangle at X(13) are all equal to 120° (Case 2), or 60°, 60°, 120° (Case 1).
The (4, 5, 6) triangle and its multiples are the only triangles with one angle being twice another and having integer sides in arithmetic progression. [39] The (3, 5, 7) triangle and its multiples are the only triangles with a 120° angle and having integer sides in arithmetic progression. [39]
The angles made by each pair of arcs at the corners of a Reuleaux triangle are all equal to 120°. This is the sharpest possible angle at any vertex of any curve of constant width. [9] Additionally, among the curves of constant width, the Reuleaux triangle is the one with both the largest and the smallest inscribed equilateral triangles. [15]
A triangle has three internal angles, each one bounded by a pair of adjacent edges; the sum of angles of a triangle always equals a straight angle (180 degrees or π radians). The triangle is a plane figure and its interior is a planar region .
These include the Calabi triangle (a triangle with three congruent inscribed squares), [10] the golden triangle and golden gnomon (two isosceles triangles whose sides and base are in the golden ratio), [11] the 80-80-20 triangle appearing in the Langley's Adventitious Angles puzzle, [12] and the 30-30-120 triangle of the triakis triangular tiling.
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For N = 3 there are two possible cases: if the triangle formed by the given points has all angles which are less than 120 degrees, the solution is given by a Steiner point located at the Fermat point; otherwise the solution is given by the two sides of the triangle which meet on the angle with 120 or more degrees.