Does anyone know of a theorem in number theory or other branch of mathematics that would prove the following: Plot the graph of the function y(x) = x ^ 2 in the first quadrant of a Cartesian x-y coordinate system. Drop a vertical line segment parallel to the y-axis, denoted as the end vertical, such that one end point of the end vertical is y(c) = c ^ 2, and the other end point is x = c. Drop another vertical line segment, denoted as the first vertical, where one end point is y(a) = a ^ 2, and the other end point is x = a. Drop another line segment, denoted as the second vertical, where one end point is y(b) = b ^ 2, and the other end point is x = b. Note, 0 < a < b < c. Define the area under the graph of the function bounded below by the x-axis and bounded to the right be the first vertical as area A. Define the area under the graph of the function bounded below by the x- axis, bounded to the left by the second vertical and bounded to the right by the end vertical as area B. Let area A = area B. Prove that the line segment on the x-axis from 0 to a is always incommensurable with the line segment from b to c, where points b and c are integral numbers.