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
Viviani's theorem Viviani's theorem Viviani's theorem, named after Vincenzo Viviani, states that the sum of the distances from "any" interior point to the sides of an equilateral triangle equals the length of the triangle's altitude. This proof depends on the readily-proved proposition that the area of a triangle is half its base times its height—that is, half the product of one side with the altitude from that side. Let ABC be an equilateral triangle whose height is "h" and whose side is "a". Let P be any point inside the triangle, and "u, s, t" the distances of P from the sides.