De Gua's theorem

De Gua's theorem states that in a tri-rectangular tetrahedron, the square of the area of the face opposite the right-angled vertex equals the sum of the squares of the other three face areas.

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Tri-rectangular tetrahedron: A tetrahedron with three mutually perpendicular edges meeting at one vertex. The theorem applies specifically when these three edges meet at right angles.
Tetrahedron: A polyhedron with four triangular faces, six edges, and four vertices. De Gua's theorem concerns a special tetrahedron with four triangular faces.
Jean-Paul de Gua de Malves: A French mathematician and scholar who lived from 1713 to 1785. The theorem bears his name and is associated with his work on geometry.
Cartesian coordinate system: A system that locates points using numerical coordinates along perpendicular axes. Placing the right-angle vertex at the origin makes the theorem's geometry explicit.
Pythagorean theorem for volumes: A family of results relating squared volumes of orthogonal components to the squared volume of a whole. These results extend the same sum-of-squares pattern to volume rather than face area.
Pythagorean theorem: A Euclidean geometry theorem stating that a right triangle's hypotenuse squared equals the sum of its legs squared. De Gua's theorem extends this sum-of-squares pattern from lengths to face areas.
Face (geometry): A flat polygonal surface forming part of the boundary of a three-dimensional solid. The theorem compares the areas of all four faces.
Euclid: An ancient Greek mathematician whose treatise Elements organized foundational results in geometry. The theorem extends a familiar Euclidean sum-of-squares relationship to tetrahedral face areas.
Analytic geometry: The study of geometric objects using coordinates, equations, and algebraic methods. Coordinates provide a concise proof by expressing each triangular face area algebraically.
Law of cosines: A relation connecting the sides of a triangle to the cosine of one of its angles. It handles non-right triangular geometry, whereas De Gua's theorem requires three perpendicular edges.