Pythagorean Theorem, Law of Sines and Law of Cosines: alternative proofs via shape derivatives

TL;DR

This paper introduces a unified method using shape derivatives to prove fundamental geometric theorems like the Pythagorean theorem, law of sines, and law of cosines, applicable in various dimensions.

Contribution

It presents a novel, simple approach based on shape derivatives to prove key triangle theorems, extending their proofs to higher dimensions.

Findings

Unified proof technique for classical theorems

Extension of proofs to higher dimensions

Simplification of geometric proofs

Abstract

We provide an alternative unified approach for proving the Pythagorean theorem (in dimension $2$ and higher), the law of sines and the law of cosines, based on the concept of shape derivative. The idea behind the proofs is very simple: we translate a triangle along a specific direction and compute the resulting change in area. Equating the change in area to zero yields the statements of the three aforementioned theorems.