Triangle inequality ≥ b ≥ c" and "formula_8" is the golden ratio, as In the case of right triangles, the triangle inequality specializes to the statement that the hypotenuse is greater than either of the two sides, and less than their sum. The second part of this theorem is already established above for any side of any triangle. The first part is established using the lower figure. In the figure, consider the right triangle . An isosceles triangle is constructed with equal sides . From the triangle postulate, the angles in the right triangle satisfy: Likewise, in the isosceles triangle , the
Fermat's right triangle theorem and the other is odd. Therefore, both formula_39 and formula_40 are even, one of which is divisible by 4. From these two numbers Fermat derives two more numbers formula_41 and formula_42, one of which is even by the previous sentence. Because formula_43 is a square, formula_44 and formula_45 are the legs of another primitive Pythagorean triangle whose area formula_46. Since formula_27 is itself a square and since formula_48 is even, formula_49 is a square. Thus, any Pythagorean triangle with square area leads to a smaller Pythagorean triangle with square area, completing the proof. Fermat's right triangle theorem Fermat's right triangle