Pythagorean theorem straightedge and compass. Pythagoras's theorem enables construction of incommensurable lengths because the hypotenuse of a triangle is related to the sides by the square root operation. The figure on the right shows how to construct line segments whose lengths are in the ratio of the square root of any positive integer. Each triangle has a side (labeled "1") that is the chosen unit for measurement. In each right triangle, Pythagoras's theorem establishes the length of the hypotenuse in terms of this unit. If a hypotenuse is related to the unit by the square root of a positive integer that is
Automedian triangle always odd, all the sides "a, b, c" have to be odd. This fact allows automedian triples to have sides and perimeter of prime numbers only. For example, (13, 17, 7) has perimeter 37. Because in a primitive automedian triangle side "a" is the sum of two squares and equal to the hypotenuse of the generating primitive Pythagorean triple, it is divisible only by primes congruent to 1 (mod 4). Consequently, "a" must be congruent to 1 (mod 4). Similarly, because the sides are related by formula_18, each of the sides "b" and "c" in the primitive automedian is the