Disphenoid

A disphenoid, also known as an equifacial or isosceles tetrahedron, is a tetrahedron whose four triangular faces are congruent. Equivalently, the tetrahedron has its three pairs of opposite edges equal in length: AB = CD, AC = BD, and AD = BC. Consequently, all four faces are triangles with the same side-length triple (p, q, r), where p, q, and r are the lengths of the three pairs of opposite edges.

The symmetry of a disphenoid is high: the lines joining the midpoints of opposite edges are pairwise perpendicular and meet at a single center. This center is equidistant from all vertices and from all faces, so the disphenoid is both cyclic (having a circumscribed sphere through the four vertices) and tangential (possessing an inscribed sphere tangent to all faces).

A disphenoid can be described by the three numbers p, q, and r, the lengths of its opposite-edge pairs. The four faces are the congruent triangles with side lengths p, q, and r; conversely, any tetrahedron with AB = CD, AC = BD, and AD = BC arises in this way. This structure makes disphenoids a notable subclass of tetrahedra, often referred to as equifacial due to their identical faces.

In summary, a disphenoid is a tetrahedron with four congruent faces and opposite edges of equal length, possessing both circumscribed and inscribed spheres with a common center and a characteristic set of three perpendicular edge-midpoint axes. It is a standard example of an equifacial (isosceles) tetrahedron.