Cosines and Cayley, Triangles and Tetrahedra

This article surveys some of the more aesthetically appealing and useful formulas relating distances, areas, and angles in triangles and tetrahedra. For example, a somewhat neglected trigonometric identity involving only the cosines of a triangle is an instance of the famous Cayley cubic surface. While most of these formulas are well known, some novel identities also make an appearance. Heron’s formula and the Cayley–Menger determinant. Many people learn in school that for a triangle with vertices A, B, and C , the area of the triangle, ABC, can be computed from Heron’s formula [2, 13], ABC = √ s(s − rAB)(s − rAC)(s − rBC) (1) in which ri j is the distance between vertex i and vertex j , and s is the semiperimeter s = rAB + rAC + rBC 2 . Almost two thousand years later, a form of Heron’s formula was found that generalizes to simplices of any dimension. This is the Cayley–Menger determinant; for a triangle: −16 ABC = det ⎛ ⎜⎜⎜⎝ 0 1 1 1 1 0 r 2 AB r 2 AC 1 r 2 AB 0 r 2 BC 1 r 2 AC r 2 BC 0 ⎞ ⎟⎟⎟⎠ = −(−rAB + rAC + rBC)(rAB − rAC + rBC)(rAB + rAC − rBC) × (rAB + rAC + rBC). The matrix in the determinant is called the Cayley–Menger matrix. Cayley found the determinantal form [7]—the polynomial itself was known earlier, by Lagrange. Menger discovered a number of further properties of the matrix and closely related variants of it [14]. For an n − 1-dimensional simplex of n vertices A1, . . . An , the volume formula generalizes to 2A1...An = (−1)n+1 2n(n!)2 det ⎛ ⎜⎜⎜⎜⎝ 0 1 1 . . . 1 1 0 r 2 A1 A2 . . . r 2 A1 An 1 r 2 A1 A2 0 . . . r 2 A2 An . . . . . . . . . . . . . . . 1 r 2 A1 An r 2 A2 An . . . 0 ⎞ ⎟⎟⎟⎟⎠ . (2) The definitive article about the Cayley–Menger matrix is [5], and its properties are also nicely summarized in [4]. The use of distances as coordinates is masterfully http://dx.doi.org/10.4169/amer.math.monthly.121.10.937 MSC: Primary 51K05, Secondary 52-02 December 2014] NOTES 937 investigated in [1] and, with particular attention paid to triangles, in [8]. Some uses and variants of the Cayley–Menger determinant are discussed in [11, 12]. Laws of cosines. For brevity in some of the more complex formulae, we write cABC for the cosine of the angle ABC. The classic law of cosines is r 2 AB − r 2 AC − r 2 BC + 2rACrBCcACB = 0. One derivation of the law of cosines makes use of the relations of the form rAB = rACcBAC + rBCcABC (3) which can be used for many purposes because of their linearity in the distances and cosines. These relations generalize to the n-dimensional simplex through the polyhedral version of the divergence theorem. If we choose a vector field to be the outward-pointing normal ni to the i th facet of a bounded convex polyhedron, then the divergence of the field is 0. This is equal to the surface integral