Commentary on fundamental properties of triangles

Commentary on fundamental properties of triangles and tetrahedrons Oscar Chavoya Aceves August 5, 2017 Abstract In this commentary, we consider to identities involving the internal angles of a triangle and the internal dihedral angles of a tetrahedron. Though these identities are no new [1], to the best of our knowledge, our interpretation of them in connection to the Pythagorean theorem is original.

Figure 1: Triangle ∆ABC Consider a triangle ∆ABC as shown in Fig. 1. The cosine function cos θ is defined as the ratio of the length ` of a segment to the length d of its orthogonal projection on a given line or segment. This ratio is considered positive when θ is an acute angle and negative when θ is obtuse. If θ is a right angle, then cos θ is expected to be zero. From these considerations it is not difficult to figure out that −a + b · cos C + c · cos B = 0

(1)

a · cos C − b + c · cos A = 0

(2)

a · cos B + b · cos A − c = 0

(3)

First, we consider this as a system of linear equations for the variables a, b and c.

−1 cos C cos B

cos C −1 cos A

1

cos B cos A −1

≡0

(4)

Which translates to cos2 A + cos2 B + cos2 C + 2 cos A cos B cos C = 1

(5)

This relation must be true for any three angles that add up to 180◦ . If one of the angles, let us say C is a right angle, then cos C = 0 and (5) is transformed into cos2 A + cos2 B = 1, (6) which is equivalent to the Pythagorean theorem a2 + b2 = c2

(7)

Figure 2: Tetrahedron ABCD We will work an analogous problem with the faces of a tetrahedron with vertexes ABCD. This figure has six edges and, correspondingly, six dihedral angles γAB = γBA , γBC = γCB , γAC = γCA , γAD = γDA , γBD = γDB , γCD = γDC , one per edge. We use the symbols ∆ABC , ∆ABD , ∆BCD , and ∆CAD to represent the areas of the corresponding triangles. The following identities follow from the principle of additivity of areas ∆ABC = ∆ABD · cos γAB + ∆BCD · cos γBC + ∆CAD · cos γCA

(8)

∆ABD = ∆ABC · cos γAB + ∆BCD · cos γBD + ∆CAD · cos γAD

(9)

∆BCD = ∆ABC · cos γBC + ∆ABD · cos γBD + ∆CAD · cos γCD

(10)

∆CAD = ∆ABC · cos γCA + ∆ABD · cos γAD + ∆BCD · cos γCD

(11)

Those equations make an homogeneous system for the areas ∆ABC , ∆ABD , ∆BCD , and ∆CAD −∆ABC + ∆ABD cos γAB + ∆BCD cos γBC + ∆CAD cos γCA = 0

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∆ABC cos γAB − ∆ABD + ∆BCD cos γBD + ∆CAD cos γAD = 0 ∆ABC cos γBC + ∆ABD cos γBD − ∆BCD + ∆CAD cos γCD = 0 ∆ABC cos γCA + ∆ABD cos γAD + ∆BCD cos γCD − ∆CAD = 0 As before, if the tetrahedron exists, the corresponding determinant must be zero −1 cos γAB cos γBC cos γCA −1 cos γBD cos γAD cos γAB (12) ≡0 cos γ cos γBD −1 cos γCD BC cos γCA cos γAD cos γCD −1 cos2 γAB + cos2 γBC + cos2 γCA + cos2 γBD + cos2 γAD + cos2 γCD (13) − cos2 γAB cos2 γCD − cos2 γBC cos2 γAD − cos2 γCA cos2 γBD +2 cos γBD cos γAD cos γCD + 2 cos γBC cos γCA cos γCD +2 cos γAB cos γCA cos γAD + 2 cos γAB cos γBC cos γBD +2 cos γAB cos γBC cos γAD cos γCD + 2 cos γAB cos γCA cos γBD cos γCD +2 cos γBC cos γCA cos γBD cos γAD = 1 This identity is the three-dimensional version of (5). In case that γAD = γBD = γCD = 90◦ it simplifies to cos2 γBD + cos2 γAD + cos2 γCD = 1 Multiplying both sides of this equations by ∆ABC 2 we get the threedimensional generalization of the Pythagorean theorem ∆ABD 2 + ∆BCD 2 + ∆CAD 2 = ∆ABC 2

(14)

Equation (5) can be easily proved to be a necessary consequence of the fact that the sum of the interior angles of a triangle equals two right angles—by the substitution C = π − A − B. The Pythagorean theorem is a necessary consequence of (5) because for the definition of the cosine function we have only used the concepts of orthogonal projection and additivity of length, which is interesting in and of itself since known proofs of the Pythagorean theorem are based on additivity of area. Equation (13) can be considered as the corresponding analogous relation in three dimensions. The sum of the measures of the interior dihedral angles of a tetrahedron is not a constant, but (13) is true. The three-dimensional Pythagorean theorem—for areas—and the result that areas can be consided as cartesian vectors is obtained as an analogous particular case.

References [1] Marshall Hampton; Cosines and Cayley, Triangles and Tetrahedra; http://www.d.umn.edu/ mhampton/MA-AMMT140030.pdf

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