LeanGeo: Formalizing Competitional Geometry problems in Lean
Chendong Song, Zihan Wang, Frederick Pu, Haiming Wang, Xiaohan Lin, Junqi Liu, Jia Li, Zhengying Liu
cs.AI
Aug 20, 2025 · v1
TL;DR
Introduces LeanGeo, a Lean 4 formal system and theorem library for competition geometry integrated with Mathlib, plus the LeanGeo-Bench benchmark.
Abstract
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 foundational 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. We open source the theorem library and the benchmark of LeanGeo at
https://github.com/project-numina/LeanGeo/tree/master.
Problem
Geometry problems are a key testbed for AI reasoning, but most existing geometry solving systems cannot express problems within a unified formal framework, making integration with other mathematical fields difficult and proof verification challenging.
Approach
The authors introduce LeanGeo, a formal system for formalizing and solving competition-level geometry problems in Lean 4. LeanGeo features a comprehensive library of high-level geometric theorems built on Lean's foundational logic, enabling rigorous proof verification and integration with Mathlib. They also present LeanGeo-Bench, a formal geometry benchmark comprising IMO and other advanced problems.
Results
Evaluation demonstrates capabilities and limitations of state-of-the-art LLMs on LeanGeo-Bench, highlighting the need for further advancements in automated geometric reasoning. The theorem library and benchmark are open-sourced.
