Theorems of Euclidean Geometry through Calculus

TL;DR

This paper presents a novel calculus-based approach to derive classical Euclidean geometry theorems by formulating and solving differential equations that relate geometric quantities.

Contribution

It introduces a method to re-derive key geometric theorems using differential equations, connecting calculus with classical geometry in a systematic way.

Findings

Successfully derives multiple classical theorems via differential equations.

Establishes an equivalence between polynomial equations and PDEs in geometric contexts.

Provides insights into relations between metric quantities through differential equations.

Abstract

We re-derive Thales, Pythagoras, Apollonius, Stewart, Heron, al Kashi, de Gua, Terquem, Ptolemy, Brahmagupta and Euler's theorems as well as the inscribed angle theorem, the law of sines, the circumradius, inradius and some angle bisector formulae, by assuming the existence of an unknown relation between the geometric quantities at stake, observing how the relation behaves under small deviations of those quantities, and naturally establishing differential equations that we integrate out. Applying the general solution to some specific situation gives a particular solution corresponding to the expected theorem. We also establish an equivalence between a polynomial equation and a set of partial differential equations. We finally comment on a differential equation which arises after a small scale transformation and should concern all relations between metric quantities.