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
Pythagorean theorem of two legs instead of the hypotenuse (see #Similar figures on the three sides). In Einstein's proof, the shape that includes the hypotenuse is the right triangle itself. The dissection consists of dropping a perpendicular from the vertex of the right angle of the triangle to the hypotenuse, thus splitting the whole triangle into two parts. Those two parts have the same shape as the original right triangle, and have the legs of the original triangle as their hypotenuses, and the sum of their areas is that of the original triangle. Because the ratio of the area of a right