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The augmented dodecahedron is Johnson solid J_(58), which is formed from augmentation of one of the faces of the regular dodecahedron by an equilateral pentagonal pyramid. It ...

Four circles c_1, c_2, c_3, and c_4 are tangent to a fifth circle or a straight line iff T_(12)T_(34)+/-T_(13)T_(42)+/-T_(14)T_(23)=0. (1) where T_(ij) is the length of a ...

The elongated pentagonal gyrobicupola is a convex equilateral solid that is Johnson solid J_(39). The unit elongated pentagonal gyrobicupola has volume ...

An elongated square pyramid is a solid formed by attaching a square pyramid atop a square prism. Such a solid might also be termed an obelisk (although that term is also used ...

A tetradecahedron is a 14-sided polyhedron, sometimes called a tetrakaidecahedron. Examples are illustrated above and summarized in the following table. name family augmented ...

If two similar figures lie in the plane but do not have parallel sides (i.e., they are similar but not homothetic), there exists a center of similitude, also called a ...

The radius of an excircle. Let a triangle have exradius r_A (sometimes denoted rho_A), opposite side of length a and angle A, area Delta, and semiperimeter s. Then r_1 = ...

The exterior angle bisectors (Johnson 1929, p. 149), also called the external angle bisectors (Kimberling 1998, pp. 18-19), of a triangle DeltaABC are the lines bisecting the ...

Let a convex polygon be inscribed in a circle and divided into triangles from diagonals from one polygon vertex. The sum of the radii of the circles inscribed in these ...

A (general) octahedron is a polyhedron having eight faces. Examples include the 4-trapezohedron, augmented triangular prism (Johnson solid J_(49)), bislit cube, Dürer solid, ...