VeriGeo: Controllable Geometry Question Generation with Numerical and Analytical Verification

2026-06-12

AI Summary

This work addresses the challenge of inconsistency among problem statements, diagrams, constraints, and solutions in automatic geometry problem generation by proposing a controllable generation framework grounded in executable reasoning traces. The framework employs collaborative Author and Solver agents to generate aligned problem-solution pairs and introduces, for the first time, a three-stage verification mechanism that integrates numerical consistency, analytical feasibility, and global coherence checks. Invalid samples are either filtered or corrected through a verification-guided reflection-and-repair strategy. Experimental results demonstrate that the proposed method significantly reduces invalid generations across five large language models. Furthermore, models fine-tuned on the 8.7k-sample VeriGeo dataset achieve state-of-the-art or competitive performance on GeoQA, PGPS9K, and MathVista-GPS benchmarks.

Abstract

Geometry problem generation is useful for AI-assisted education and multimodal mathematical reasoning, but reliable synthesis remains difficult because the problem statement, diagram, constraints, and solution should be mutually consistent. Existing methods often trade off controllability and reliability: seed-based rewriting is flexible but weakly verifiable, whereas diagram-first construction improves validity but is less suited to arbitrary user-specified constraints. We introduce VeriGeo, a controllable geometry generation framework grounded in executable reasoning traces. Given user constraints such as target concepts and difficulty, an Author agent generates a problem and diagram, and a Solver agent produces a proof-aligned solution. Both agents use a shared action sequence that connects natural language, diagrams, geometric constraints, and proof steps into a verifiable representation. A three-stage pipeline checks numerical consistency, analytical realizability, and global consistency, using verification-guided reflection to repair recoverable failures and reject unrecoverable ones. Across five LLM backbones, raw generations frequently fail these checks, while VeriGeo repairs a substantial fraction of the invalid attempts. Supervised fine-tuning on 8.7k examples generated by VeriGeo achieves the best reported GeoQA performance among end-to-end multimodal LLM-based solvers, and obtains strong results on PGPS9K and MathVista-GPS, demonstrating the effectiveness of verified synthetic data for improving multimodal geometry reasoning.

Problem

• geometry question generation
• controllability
• reliability
• consistency verification
• multimodal reasoning

Innovation

• controllable generation
• executable reasoning traces
• verification-guided reflection
• multimodal geometry reasoning
• synthetic data verification