Constructions by ruler and compass, together with a fixed conic

TL;DR

This paper proves that any point constructible with conics can be constructed using only a ruler, compass, and a single fixed conic, simplifying the classical tools needed for such constructions.

Contribution

It demonstrates that a single fixed conic, different from a circle, suffices to perform all constructions previously requiring multiple conics.

Findings

Any point constructible from conics can be constructed with a ruler, compass, and one fixed conic.

The result simplifies classical constructions by reducing the number of conics needed.

The proof bridges classical geometric constructions with conic-based methods.

Abstract

It is well-known to be impossible to trisect an arbitrary angle and duplicate an arbitrary cube by a ruler and a compass. On the other hand, it is known from the ancient times that these constructions can be performed when it is allowed to use several conic curves. In this paper, we prove that any point constructible from conics can be constructed using a ruler and a compass, together with a single fixed non-degenerate conic different from a circle.