Pythagorean conics and homothetic triangles

TL;DR

The paper generalizes the Pythagorean theorem to the lengths of conic arcs built on the sides of a right triangle, proving the relation $c_1^2 = c_2^2 + c_3^2$ when the conic eccentricity is fixed and the sagitta-to-side ratio $k$ is constant. The authors develop both analytical and synthetic proofs, leveraging the arc-length formula in polar coordinates and showing that each arc length satisfies $c_i = g(k) l_i$, where $g(k)$ is common to all three sides. They establish existence and uniqueness of the conic arcs, identify a common angle subtended by each side, and reveal a homothety between the original and enveloping triangles, with the centre of homothety termed the Pythagorean centre. This framework yields an infinite family of Pythagorean conic triples and highlights a constructive geometric pathway to generate new triples without heavy computation. The work opens avenues for extending the result to other curve families and non-Euclidean geometries, and suggests potential applications of the Pythagorean centre in geometric design and analysis.

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.

Figures (8)

• Figure 1: Conic arc constructed symmetrically on each side of a right triangle.
• Figure 2: Uniqueness of the circular arc under the specified conditions.
• Figure 3: Construction of the unique parabola under the given conditions.
• Figure 4: Elliptic arc under the specified conditions.
• Figure 5: Elliptic arc symmetrically constructed on a segment of length $l$.

Theorems & Definitions (3)

• Lemma 2.1
• Lemma 2.2
• Theorem 2.1