Given a square whose diagonal is equal to one:
or
Given an isosceles right triangle whose hypotenuse is equal to one:
Note that the triangle is 1/2 the square. Any relationship found between the side and hypotenuse of the triangle is the same for the side and diagonal of the square. [Hypotenuse, H, = Diagonal D, = 1].
From the point, B, on the hypotenuse, using the side BC as radius, mark off the point, D, on the hypotenuse. [BC = AC = BD = x].
From D, drop a perpendicular intersecting BC, at E. [DE = BE = y].
Then triangle DBE is similar to triangle ABC and their corresponding parts are proportional.
Therefore:
The side of the small triangle, DE = y, is to the side of the large triangle, AC
= x, as the hypotenuse of the small triangle, DB = x, is to the hypotenuse, AB =
1, of the large triangle.
i.e., y : x :: x : y, or y/x = x/1 or y = x²
Curious!
An isosceles right triangle whose hypotenuse equals x, and whose side equals x squared! By completing the square, it has the same relationship: its side is the square of the diagonal.
If there is no common measure between the side and diagonal of the square, then there is no assurance that a number and its square have a common measure either.
Curious.