User:GianniG46/sandbox2

Proof by dimensional analysis
A very simple proof of the Pythagorean theorem can be obtained by dimensional analysis. The area of a triangle depends on its size and shape, which can be unambiguously identified by the length of one of its edges (for example, the largest) and by any two of its angles (the third being determined by the fact that the sum of all three is π). Thus, recalling that an area has the dimensions of a length squared, we can write:


 * area = largest_edge² • f (angle_1, angle_2),

where f is an adimensional function of the angles. Now, referring to the figure at right, if we divide a right triangle into two smaller ones by tracing the segment perpendicular to its hypotenuse and passing by the opposite vertex, and express the obvious fact that the total area is the sum of the two smaller areas, by applying the previos equation we have:


 * a² • f (α, π/2) = b² • f (α, π/2) + c² • f (α, π/2).

And, eliminating f:


 *  a² = b² + c² , Q.E.D.

Note that the result is obtained without specifying the form of the adimensional function f.