A Group Theory Proof of Pascal's Theorem

TL;DR

This paper presents a new proof of Pascal's Theorem by linking it to the associativity of a natural binary operation on conic sections, demonstrating the theorem through algebraic and geometric equivalences.

Contribution

It introduces a novel algebraic approach to Pascal's Theorem by establishing its equivalence to the associativity of a specific binary operation on conics, independent of the theorem itself.

Findings

Pascal's Theorem is equivalent to the associativity of a binary operation on conics.

The binary operation corresponds to addition, multiplication, or rotation depending on the conic type.

The proof leverages known associative properties of real number operations and rotations.

Abstract

It will be shown that Pascal's Theorem is equivalent to the associativity of a natural binary operation on conic sections. A novel proof for Pascal's Theorem will then be given by showing that this binary operation is associative independent of Pascal's Theorem. Specifically, this operation is equivalent to either addition of real numbers, multiplication of real numbers, or rotations on the plane depending on the type of the conic. Since each of these is already known to be associative, it will follow that the binary operation is associative and this will prove Pascal's Theorem.