Bisecting horn angles

TL;DR

This paper explores the classical geometric problem of bisecting horn angles, providing an ancient construction method and comparing it to modern conformal bisection techniques.

Contribution

It presents a historical geometric construction for bisecting horn angles and analyzes its relation to contemporary conformal bisection methods.

Findings

Constructed a bisector for horn angles using ancient geometric tools.

Compared ancient construction with modern conformal bisection methods.

Demonstrated the feasibility of ancient methods within classical geometry.

Abstract

A horn angle between a circle and its tangent is considered in Euclid's Elements, and Euclid remarks that it is smaller than any acute rectilinear angle. Already in antiquity, Proclus wondered whether it is possible to bisect horn angles. We will give a construction of a bisector which was within the means of ancient geometers since the time of Archimedes and Apollonius. We will also compare it to the conformal bisection method introduced in modern times.