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
Group of rational points on the unit circle to the two most famous Pythagorean right triangles: 3,4,5 and 5,12,13) are elements of "G", and their group product is (−33/65 + i56/65), which corresponds to a 33,56,65 Pythagorean right triangle. The sum of the squares of the numerators 33 and 56 is 1089 + 3136 = 4225, which is the square of the denominator 65. The set of all 2×2 rotation matrices with rational entries coincides with G.This follows from the fact that the circle group formula_2 is isomorphic to formula_3, and the fact that their rational points coincide. The structure of "G" is an infinite sum of cyclic