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 this equation has no solution, and that this case of Fermat's last theorem is true. Another equivalent formulation of the same problem involves congruent numbers, the numbers that are areas of right triangles whose three sides are all rational numbers. By multiplying the sides by a common denominator, any congruent number may be transformed into the area of a Pythagorean triangle, from which it follows that the congruent numbers are exactly the numbers formed by multiplying a congruum by the square of a rational number. Thus, there is no square congruum if and only if the number 1 is not