quality:correction, structural:scholar_edition
Proof using differentials

One can arrive at the Pythagorean theorem by studying how changes in a side produce a change in the hypotenuse and employing calculus.

The triangle ABC is a right triangle, as shown in the upper part of the diagram, with BC the hypotenuse. At the same time the triangle lengths are measured as shown, with the hypotenuse of length y, the side AC of length x and the side AB of length a, as seen in the lower diagram part.

If x is increased by a small amount dx by extending the side AC slightly to D, then y also increases by dy. These form two sides of a triangle, CDE, which (with E chosen so CE is perpendicular to the hypotenuse) is a right triangle approximately similar to ABC. Therefore, the ratios of their sides must be the same, that is:

This can be rewritten as y d y = x d x {\displaystyle y\,dy=x\,dx}, which is a differential equation that can be solved by direct integration:

giving

The constant can be deduced from x = 0, y = a to give the equation

This is more of an intuitive proof than a formal one: it can be made more rigorous if proper limits are used in place of dx and dy.