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
Automedian triangle always odd, all the sides "a, b, c" have to be odd. This fact allows automedian triples to have sides and perimeter of prime numbers only. For example, (13, 17, 7) has perimeter 37. Because in a primitive automedian triangle side "a" is the sum of two squares and equal to the hypotenuse of the generating primitive Pythagorean triple, it is divisible only by primes congruent to 1 (mod 4). Consequently, "a" must be congruent to 1 (mod 4). Similarly, because the sides are related by formula_18, each of the sides "b" and "c" in the primitive automedian is the