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
Fermat's right triangle theorem single Diophantine equation which can be simplified to Conversely, any solution to this equation could be factored to give a square congruum. (In particular, the squares formula_13, formula_14, and formula_15 form an arithmetic progression with congruum formula_16, which is a square itself.) Thus, the solvability of this equation is equivalent to the existence of a square congruum. But, if Fermat's last theorem were false for the exponent formula_17, then squaring one of the three numbers in any counterexample would also give three numbers that solve this equation. Therefore, Fermat's proof that no Pythagorean triangle has a square area implies that