LeanGeo: Formalizing Competitional Geometry problems in Lean

Geometry problems are a crucial testbed for AI reasoning capabilities. Most existing geometry solving systems cannot express problems within a unified framework, thus are difficult to integrate with other mathematical fields. Besides, since most geometric proofs rely on intuitive diagrams, verifying geometry problems is particularly challenging. To address these gaps, we introduce LeanGeo, a unified formal system for formalizing and solving competition-level geometry problems within the Lean 4 theorem prover. LeanGeo features a comprehensive library of high-level geometric theorems with Lean's founda-tional logic, enabling rigorous proof verification and seamless integration with Mathlib. We also present LeanGeo-Bench, a formal geometry benchmark in LeanGeo, comprising problems from the International Mathematical Olympiad (IMO) and other advanced sources. Our evaluation demonstrates the capabilities and limitations of state-of-the-art Large Language Models on this benchmark, highlighting the need for further advancements in automated geometric reasoning. In recent years, Large Language Models (LLMs) have made significant progress in mathematical reasoning, particularly in automated theorem proving (Bibel, 2013). Formal theorem proving is a crucial domain for ensuring the correctness of hard-to-verify proofs within theorem proving. Lean 4 (Moura & Ullrich, 2021), as a prominent proof assistant, provides a solid foundation for algebra and number theory through its extensive Mathlib library (mathlib community, 2020). It has been widely used in the formal verification of theorems within LLMs. However, Euclidean geometry, an essential component of mathematical reasoning and a frequent focus of competitions, remains relatively underexplored in Lean 4 community, Mathlib and automated theorem provers.