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
Pythagorean prime representation can be interpreted geometrically: the Pythagorean primes are exactly the odd prime numbers "p" such that there exists a right triangle, with integer legs, whose hypotenuse has length . They are also exactly the prime numbers "p" such that there exists a right triangle with integer sides whose hypotenuse has length "p". For, if the triangle with legs "x" and "y" has hypotenuse length (with "x" > "y"), then the triangle with legs "x" − "y" and 2"xy" has hypotenuse length "p". Another way to understand this representation as a sum of two squares involves Gaussian integers, the complex