Pythagorean conics and homothetic triangles

TL;DR

This paper generalizes the Pythagorean theorem to conic arcs on right triangles, establishing conditions for its validity and identifying a new Pythagorean centre related to homothetic triangles.

Contribution

It introduces a novel generalization of the Pythagorean theorem involving conic arcs and defines a new Pythagorean centre based on homothety concepts.

Findings

The theorem holds for fixed conic eccentricity with constant arc-to-side ratio.

Existence of a Pythagorean centre related to homothetic triangles.

The generalization applies to an infinite set of conics derived from right triangles.

Abstract

This study investigates a generalisation of the Pythagorean theorem to the lengths of conic arcs constructed symmetrically on the sides of a right triangle. It is demonstrated that the theorem remains valid whenever the conic eccentricity is fixed and the ratio between the length of each arc sagitta and its corresponding side is constant. We identify the existence of a Pythagorean centre, defined as the common centre of all homotheties between the original right triangle and the enveloping right triangles of the infinite set of Pythagorean triples of conics that can be derived from it. The proofs rely on the application of both geometrical and analytical techniques starting from classical Pythagoras theorem.