Gold-Medal-Level Olympiad Geometry Solving with Efficient Heuristic Auxiliary Constructions

TL;DR

This paper introduces HAGeo, a CPU-based heuristic method for automated geometry theorem proving that achieves medal-level performance on IMO problems, surpassing neural network approaches and establishing a new challenging benchmark.

Contribution

The paper presents HAGeo, a novel heuristic auxiliary construction method that significantly improves automated geometry theorem proving performance on IMO-level problems.

Findings

HAGeo solves 28 of 30 IMO-30 problems, achieving gold-medal level performance.

HAGeo outperforms AlphaGeometry, a neural network-based approach.

HAGeo-409 benchmark offers a more challenging and comprehensive evaluation environment.

Abstract

Automated theorem proving in Euclidean geometry, particularly for International Mathematical Olympiad (IMO) level problems, remains a major challenge and an important research focus in Artificial Intelligence. In this paper, we present a highly efficient method for geometry theorem proving that runs entirely on CPUs without relying on neural network-based inference. Our initial study shows that a simple random strategy for adding auxiliary points can achieve silver-medal level human performance on IMO. Building on this, we propose HAGeo, a Heuristic-based method for adding Auxiliary constructions in Geometric deduction that solves 28 of 30 problems on the IMO-30 benchmark, achieving gold-medal level performance and surpassing AlphaGeometry, a competitive neural network-based approach, by a notable margin. To evaluate our method and existing approaches more comprehensively, we further construct HAGeo-409, a benchmark consisting of 409 geometry problems with human-assessed difficulty levels. Compared with the widely used IMO-30, our benchmark poses greater challenges and provides a more precise evaluation, setting a higher bar for geometry theorem proving.