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" and HP" (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" and HH" (n >~ 2).
1. PRELIMINARIES
Let be K~{C, H}, n ~> 2. We describe points in KP" as one-dimensional subspaces of the right-vector space K "+ 1. For x ~ K "+ 1(0} let [x] .'= x. K denote the corresponding point in KP", thus [x] = Ix" 2] for all 2 ~ K{0}. n+l (x, y) ,= ~ ~iy~, i=1 thus (x2, y#) = ~-(x, y)p and (y, x) = (x, y). II II denotes the norm, 121 the absolute value, Re2 the real part, IIm,~l := 12 -Re 21 the absolute value of the imaginary part of 2 e K. The metric d on KP" is defined by I(x,y)l d(I-x], fy-1).'= arccos II x I1" II Y In the following let Ix], [y], [z] eKP" with Ilxll = [lyll = Ilzll = 1. DEFINITION. A triangle is a triple (Ix], I-y], [z]) of three different points in KP" such that the distance between any two of them is strictly smaller than re/2.
We imagine the vertices [x], I-y], I-z] of a triangle as being connected by the (uniquely existing) shortest paths between them, called sides (Figure 1). Since our metric is an inner metric we have that the length of the side