A Machine-Checked Direct Proof of the Steiner-Lehmus Theorem

TL;DR

This paper presents a formal, constructive proof of the Steiner-Lehmus theorem using a proof assistant, ensuring the proof is genuinely direct without reliance on indirect methods like reductio ad absurdum.

Contribution

It provides the first formal, constructive, and verified direct proof of the Steiner-Lehmus theorem within a proof assistant based on constructive logic.

Findings

Proof is formalized in a constructive axiom set

The proof is verified by a proof assistant

Ensures the proof is genuinely direct without indirect methods

Abstract

A direct proof of the Steiner-Lehmus theorem has eluded geometers for over 170 years. The challenge has been that a proof is only considered direct if it does not rely on reductio ad absurdum. Thus, any proof that claims to be direct must show, going back to the axioms, that all of the auxiliary theorems used are also proved directly. In this paper, we give a proof of the Steiner-Lehmus theorem that is guaranteed to be direct. The evidence for this claim is derived from our methodology: we have formalized a constructive axiom set for Euclidean plane geometry in a proof assistant that implements a constructive logic and have built the proof of the Steiner-Lehmus theorem on this constructive foundation.