# GeometryZero: Advancing Geometry Solving via Group Contrastive Policy Optimization

arXiv: https://arxiv.org/abs/2506.07160 · HTML: https://mlsanity.com/paper/2506.07160/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Placed in Spatial and embodied reasoning, one of 17 directions in LLM reasoning.
- Builds on 27 of its 43 references that are in the map.
- First posted on arXiv 8 Jun 2025.

## Placed in

- LLM reasoning / [Spatial and embodied reasoning](https://mlsanity.com/lr/spatial-and-embodied-reasoning/)

## Builds on (most cited first, 27 of 27)

- [Training language models to follow instructions with human feedback](https://mlsanity.com/paper/2203.02155/) (2022)
- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Direct Preference Optimization: Your Language Model is Secretly a Reward Model](https://mlsanity.com/paper/2305.18290/) (2023)
- [Efficient Memory Management for Large Language Model Serving with PagedAttention](https://mlsanity.com/paper/2309.06180/) (2023)
- [Lost in the Middle: How Language Models Use Long Contexts](https://mlsanity.com/paper/2307.03172/) (2023)
- [Gemini 1.5: Unlocking multimodal understanding across millions of tokens of context](https://mlsanity.com/paper/2403.05530/) (2024)
- [Trust Region Policy Optimization](https://mlsanity.com/paper/1502.05477/) (2015)
- [Learn to Explain: Multimodal Reasoning via Thought Chains for Science Question Answering](https://mlsanity.com/paper/2209.09513/) (2022)
- [MathVista: Evaluating Mathematical Reasoning of Foundation Models in Visual Contexts](https://mlsanity.com/paper/2310.02255/) (2023)
- [ChartQA: A Benchmark for Question Answering about Charts with Visual and Logical Reasoning](https://mlsanity.com/paper/2203.10244/) (2022)
- [OlympiadBench: A Challenging Benchmark for Promoting AGI with Olympiad-Level Bilingual Multimodal Scientific Problems](https://mlsanity.com/paper/2402.14008/) (2024)
- [Large Language Models Can Be Easily Distracted by Irrelevant Context](https://mlsanity.com/paper/2302.00093/) (2023)
- [Visual-RFT: Visual Reinforcement Fine-Tuning](https://mlsanity.com/paper/2503.01785/) (2025)
- [Video-R1: Reinforcing Video Reasoning in MLLMs](https://mlsanity.com/paper/2503.21776/) (2025)
- [L1: Controlling How Long A Reasoning Model Thinks With Reinforcement Learning](https://mlsanity.com/paper/2503.04697/) (2025)
- [Inter-GPS: Interpretable Geometry Problem Solving with Formal Language and Symbolic Reasoning](https://mlsanity.com/paper/2105.04165/) (2021)
- [Making Retrieval-Augmented Language Models Robust to Irrelevant Context](https://mlsanity.com/paper/2310.01558/) (2023)
- [Dynamic Prompt Learning via Policy Gradient for Semi-structured Mathematical Reasoning](https://mlsanity.com/paper/2209.14610/) (2022)
- [Visual Sketchpad: Sketching as a Visual Chain of Thought for Multimodal Language Models](https://mlsanity.com/paper/2406.09403/) (2024)
- [LMM-R1: Empowering 3B LMMs with Strong Reasoning Abilities Through Two-Stage Rule-Based RL](https://mlsanity.com/paper/2503.07536/) (2025)
- [GeomVerse: A Systematic Evaluation of Large Models for Geometric Reasoning](https://mlsanity.com/paper/2312.12241/) (2023)
- [Gold-medalist Performance in Solving Olympiad Geometry with AlphaGeometry2](https://mlsanity.com/paper/2502.03544/) (2025)
- [CPPO: Accelerating the Training of Group Relative Policy Optimization-Based Reasoning Models](https://mlsanity.com/paper/2503.22342/) (2025)
- [Acting Less is Reasoning More! Teaching Model to Act Efficiently](https://mlsanity.com/paper/2504.14870/) (2025)
- [PRISM: Self-Pruning Intrinsic Selection Method for Training-Free Multimodal Data Selection](https://mlsanity.com/paper/2502.12119/) (2025)
- [VisuoThink: Empowering LVLM Reasoning with Multimodal Tree Search](https://mlsanity.com/paper/2504.09130/) (2025)

## Built on by (most cited first, 9 of 9)

- [DynaSolidGeo: A Dynamic Benchmark for Genuine Spatial Mathematical Reasoning of VLMs in Solid Geometry](https://mlsanity.com/paper/2510.22340/) (2025)
- [Do Multi-Agents Solve Better Than Single? Evaluating Agentic Frameworks for Diagram-Grounded Geometry Problem Solving and Reasoning](https://mlsanity.com/paper/2512.16698/) (2025)
- [Thinking with Constructions: A Benchmark and Policy Optimization for Visual-Text Interleaved Geometric Reasoning](https://mlsanity.com/paper/2603.18662/) (2026)
- [LatentGeo: Learnable Auxiliary Constructions in Latent Space for Multimodal Geometric Reasoning](https://mlsanity.com/paper/2603.12166/) (2026)
- [Free Energy-Driven Reinforcement Learning with Adaptive Advantage Shaping for Unsupervised Reasoning in LLMs](https://mlsanity.com/paper/2605.04065/) (2026)
- [Adapt to Thrive! Adaptive Power-Mean Policy Optimization for Improved LLM Reasoning](https://mlsanity.com/paper/2605.04066/) (2026)
- [Draw2Think: Harnessing Geometry Reasoning through Constraint Engine Interaction](https://mlsanity.com/paper/2605.20743/) (2026)
- [Learning Where Outcomes Change:Credit-Addressable Reasoning for Multimodal Geometry](https://mlsanity.com/paper/2608.30457/) (2026)
- [From Symbolic Perception to Logical Deduction: A Framework for Guiding Language Models in Geometric Reasoning](https://mlsanity.com/paper/2609.10335/) (2026)

## Abstract (arXiv)

Recent progress in large language models (LLMs) has boosted mathematical reasoning, yet geometry remains challenging where auxiliary construction is often essential. Prior methods either underperform or depend on very large models (e.g., GPT-4o), making them costly. We argue that reinforcement learning with verifiable rewards (e.g., GRPO) can train smaller models to couple auxiliary construction with solid geometric reasoning. However, naively applying GRPO yields unconditional rewards, encouraging indiscriminate and sometimes harmful constructions. We propose Group Contrastive Policy Optimization (GCPO), an RL framework with two components: (1) Group Contrastive Masking, which assigns positive/negative construction rewards based on contextual utility, and (2) a Length Reward that encourages longer reasoning chains. On top of GCPO, we build GeometryZero, an affordable family of geometry reasoning models that selectively use auxiliary construction. Experiments on Geometry3K and MathVista show GeometryZero consistently outperforms RL baselines (e.g., GRPO, ToRL). The code has been available at https://github.com/ekonwang/GeometryZero.