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
Viviani's theorem Viviani's theorem Viviani's theorem, named after Vincenzo Viviani, states that the sum of the distances from "any" interior point to the sides of an equilateral triangle equals the length of the triangle's altitude. This proof depends on the readily-proved proposition that the area of a triangle is half its base times its height—that is, half the product of one side with the altitude from that side. Let ABC be an equilateral triangle whose height is "h" and whose side is "a". Let P be any point inside the triangle, and "u, s, t" the distances of P from the sides.