Gold-Medal-Level Olympiad Geometry Solving with Efficient Heuristic Auxiliary Constructions
Authors: name: Boyan Duan, name: Xiao Liang, name: Shuai Lu, name: Yaoxiang Wang, name: Yelong Shen, name: Kai-Wei Chang, name: Ying Nian Wu, name: Mao Yang, name: Weizhu Chen, name: Yeyun Gong
Organization: Microsoft

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.