Sum of angles of a triangle The relation between angular defect and the triangle's area was first proven by Johann Heinrich Lambert. One can easily see how hyperbolic geometry breaks Playfair's axiom, Proclus' axiom (the parallelism, defined as non-intersection, is intransitive in an hyperbolic plane), the equidistance postulate (the points on one side of, and equidistant from, a given line do not form a line), and Pythagoras' theorem. A circle cannot have arbitrarily small curvature, so the three points property also fails. The sum of the angles can be arbitrarily small (but positive). For an ideal triangle, a generalization of hyperbolic triangles, this sum is equal
Proofs of Fermat's theorem on sums of two squares formula_196. The area of the fundamental parallelogram (in this case a rectangle) of the lattice is formula_82. The area of the open disk, formula_198, of radius formula_199 centered around the origin is formula_200. Furthermore, formula_198 is convex and symmetrical about the origin. Therefore, by Minkowski's theorem there exists a nonzero vector formula_196 such that formula_203. Both formula_204 and formula_205 so formula_206. Hence formula_82 is the sum of the squares of the components of formula_208. Let formula_209 be prime, let formula_210 denote the natural numbers (with or without zero), and consider the finite set formula_211 of triples of numbers. Then formula_212