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The shape invariant of triangles and trigonometry in two-point homogeneous spaces

Abstract

We define a fourth basic invariant, which, besides the lengths of the three sides of a triangle, determines a triangle in the complex and quaternion projective spaces ℂP n and ℍP n (n≥2) uniquely up to isometry. We give inequalities describing the exact range of the four basic invariants. We express the angular invariants of a triangle with our basic invariants, giving a new completely elementary proof of the laws of trigonometry. As a corollary we derive a large number of congruence theorems. Finally we get, in exactly the same way, the corresponding results for triangles in the complex and quaternion hyperbolic spaces ℂH n and ℍH n (n≥2).

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Brehm, U. The shape invariant of triangles and trigonometry in two-point homogeneous spaces. Geom Dedicata 33, 59–76 (1990). https://doi.org/10.1007/BF00147601

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  • DOI: https://doi.org/10.1007/BF00147601

Keywords

  • Projective Space
  • Homogeneous Space
  • Hyperbolic Space
  • Elementary Proof
  • Basic Invariant