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A tetrahedron having a trihedron all of the face angles of which are right angles. The face opposite the vertex of the right angles is called the base. If the edge lengths bounding the trihedral angle are , , and , then the side lengths of the base are given by , , and , and so has semiperimeter
(1)
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The volume of the trirectangular tetrahedron is
(2)
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Using Heron's formula, the surface area is therefore
(3)
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Let be the area of the triangle with vertices , , and . The remarkable de Gua's theorem
(4)
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then follows from the identity
(5)
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with defined by (1).