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 that the numbers to be squared are the difference of legs, hypotenuse, and sum of legs of a Pythagorean triangle, and that the congruum is four times the area of the same triangle. In his later work on the congruum problem, published in "The Book of Squares", Fibonacci observed that it is impossible for a congruum to be a square number itself, but did not present a satisfactory proof of this fact. If three squares formula_4, formula_5, and formula_6 could form an arithmetic progression whose congruum was also a square formula_7, then these numbers would satisfy the Diophantine equations That