Equilateral triangle is the only triangle with integer sides and three rational angles as measured in degrees. The equilateral triangle is the only acute triangle that is similar to its orthic triangle (with vertices at the feet of the altitudes) (the heptagonal triangle being the only obtuse one). Equilateral triangles are found in many other geometric constructs. The intersection of circles whose centers are a radius width apart is a pair of equilateral arches, each of which can be inscribed with an equilateral triangle. They form faces of regular and uniform polyhedra. Three of the five Platonic solids are composed of equilateral
Sum of angles of a triangle Sum of angles of a triangle In several geometries, a triangle has three "vertices" and three "sides", where three angles of a triangle are formed at each vertex by a pair of adjacent sides. In a Euclidean space, the sum of measures of these three angles of any triangle is invariably equal to the straight angle, also expressed as 180 °, radians, two right angles, or a half-turn. It was unknown for a long time whether other geometries exist, where this sum is different. The influence of this problem on mathematics was particularly strong during the 19th century. Ultimately, the