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An equilateral triangle is a triangle that has three equal sides. It is a special case of an isosceles triangle in the modern definition, stating that an isosceles triangle is defined at least as having two equal sides. [1] Based on the modern definition, this leads to an equilateral triangle in which one of the three sides may be considered ...
Triangles have many types based on the length of the sides and the angles. A triangle whose sides are all the same length is an equilateral triangle, [3] a triangle with two sides having the same length is an isosceles triangle, [4] [a] and a triangle with three different-length sides is a scalene triangle. [7]
A polygon ear. One way to triangulate a simple polygon is based on the two ears theorem, as the fact that any simple polygon with at least 4 vertices without holes has at least two "ears", which are triangles with two sides being the edges of the polygon and the third one completely inside it. [5]
Given a square, construct equilateral triangles on two adjacent edges, either both inside or both outside the square. Then the triangle formed by joining the vertex of the square distant from both triangles and the vertices of the triangles distant from the square is equilateral. [2]
The 24 edges can be seen in 4 central hexagons. With octahedral symmetry (orbifold 432), the squares have the 4-fold symmetry, triangles the 3-fold symmetry, and vertices the 2-fold symmetry. With tetrahedral symmetry (orbifold 332) the 24 vertices split into 2 edge classes, and the 8 triangles split into 2 face classes. The square symmetry is ...
Alternatively, a Reuleaux triangle may be constructed from an equilateral triangle T by drawing three arcs of circles, each centered at one vertex of T and connecting the other two vertices. [9] Or, equivalently, it may be constructed as the intersection of three disks centered at the vertices of T, with radius equal to the side length of T. [10]
The relations can be made apparent by examining the vertex figures obtained by listing the faces adjacent to each vertex (remember that for uniform polyhedra all vertices are the same, that is vertex-transitive). For example, the cube has vertex figure 4.4.4, which is to say, three adjacent square faces. The possible faces are 3 - equilateral ...
Broken down, 3 6; 3 6 (both of different transitivity class), or (3 6) 2, tells us that there are 2 vertices (denoted by the superscript 2), each with 6 equilateral 3-sided polygons (triangles). With a final vertex 3 4.6, 4 more contiguous equilateral triangles and a single regular hexagon.