Euclid-Omni : A Unified Neuro-Symbolic Framework for Plane Geometry

Researchers introduced Euclid-Omni, a unified neuro-symbolic framework that combines a formal geometry system with large language models (LLMs) and vision-language models (VLMs) to solve plane geometry problems up to Olympiad-level difficulty. The framework's core component, Euclidea, is a symbolic solver that generates reasoning steps via deductive inference and algebraic computation, and a data-generation pipeline synthesizes problems, solutions, and diagrams for training. Experiments show VLMs trained on synthetic data achieve superior performance on calculation tasks, and LLMs combined with Euclidea are competitive with state-of-the-art systems on Olympiad-level proving problems while using orders of magnitude less compute and training data.

arXiv:2608.14585v1 Announce Type: new Abstract: Euclidean geometry is a compelling testbed for AI reasoning, as it demands the combination of intuitive diagram understanding, axiomatic deduction, and algebraic computation. Yet, existing approaches typically address only a subset of these abilities or struggle with competition-level problems. We introduce \textit{Euclid-Omni}, a unified neuro-symbolic framework that couples a formal geometry system with Large Language Models LLMs and Vision-Language Models VLMs to tackle both calculation- and proving-style problems, in formal and natural languages, up to Olympiad-level difficulty. At its core, we develop \textit{Euclidea}, a versatile symbolic geometry solver that automatically generates reasoning steps through deductive inference and algebraic computation. Building on this, we develop a data-generation pipeline that synthesizes symbolic problems and solutions, renders diagrams, and translates them into natural language, producing large-scale, diverse datasets for training LLMs and VLMs across a wide range of reasoning settings. Experiments show that VLMs trained on our synthetic data achieve superior performance on calculation tasks, and that LLMs combined with \textit{Euclidea} are competitive with state-of-the-art systems on Olympiad-level proving problems, despite using orders of magnitude less compute and training data. Code and scripts are publicly available at https://github.com/20171130/Euclid-Omni