On Kakeya's Geometric Proof of Enestr\"om-Kakeya's Theorem

TL;DR

This paper revisits Kakeya's 1912 geometric proof of the Enestr"om-Kakeya theorem, providing a detailed analysis of the geometric structure and offering an alternative proof based on interlacing circles.

Contribution

It explicitly demonstrates Kakeya's geometric structure and presents an equivalent proof using internally interlacing circles, which was not previously established.

Findings

Calculated centers and radii of interlacing circles.

Proved Kakeya's geometric structure.

Provided an alternative proof based on interlacing circles.

Abstract

This paper is devoted to demonstrate Kakeya's geometric proof of his theorem (1912), independently established earlier by Enestr\"om (1893). By calculating centers and radii of the interlacing circles of Kakeya's method, we prove Kakeya's geometric structure, which has not been previously established. We give an equivalent proof, which is based on the construction of internally interlacing circles, which has been geometrically considered by Tomic (1948).