# Reactivating Test-Time Scaling for Plane Geometry Problem Solving

arXiv: https://arxiv.org/abs/2608.30156 · HTML: https://mlsanity.com/paper/2608.30156/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Spatial and embodied reasoning, one of 17 directions in LLM reasoning.
- Builds on 9 of its 34 references that are in the map.
- First posted on arXiv 31 Aug 2026.

## Placed in

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

## Builds on (most cited first, 9 of 9)

- [GPT-4 Technical Report](https://mlsanity.com/paper/2303.08774/) (2023)
- [Qwen2.5-VL Technical Report](https://mlsanity.com/paper/2502.13923/) (2025)
- [Gemini 2.5: Pushing the Frontier with Advanced Reasoning, Multimodality, Long Context, and Next Generation Agentic Capabilities](https://mlsanity.com/paper/2507.06261/) (2025)
- [FormalGeo: An Extensible Formalized Framework for Olympiad Geometric Problem Solving](https://mlsanity.com/paper/2310.18021/) (2023)
- [Fuse, Reason and Verify: Geometry Problem Solving with Parsed Clauses from Diagram](https://mlsanity.com/paper/2407.07327/) (2024)
- [Milestones over Outcome: Unlocking Geometric Reasoning with Sub-Goal Verifiable Reward](https://mlsanity.com/paper/2601.05073/) (2026)
- [Bridging Formal Language with Chain-of-Thought Reasoning to Geometry Problem Solving](https://mlsanity.com/paper/2508.09099/) (2025)
- [GeoFM: Enhancing Geometric Reasoning of MLLMs via Synthetic Data Generation through Formal Language](https://mlsanity.com/paper/2510.27448/) (2025)
- [GeoThought: A Dataset for Enhancing Mathematical Geometry Reasoning in Vision-Language Models](https://mlsanity.com/paper/2510.21881/) (2025)

## Abstract (arXiv)

Plane geometry problem (PGP) solving has become a critical benchmark for multimodal reasoning because it requires accurate visual perception and precise multi-step symbolic deduction. Although test-time scaling (TTS) has demonstrated remarkable success in general mathematical reasoning, it fails to scale effectively under the symbolic-program paradigm for plane geometry. We identify two key obstacles: limited reasoning diversity induced by rigid symbolic programs and insufficient explicit visual grounding before symbolic deduction. To address these issues, we propose Multi-Trace Synthesis (MTS), which converts each symbolic program into heterogeneous reasoning traces, including executable Python scripts and CoT-augmented variants. We further propose Perception-Augmented (PA) training, which parses diagrams into structured semantic clauses before deduction, and Consensus-Guided Multi-Trace Ensemble (CG-MTE) for efficient self-adaptive inference. Experiments on three geometry benchmarks show that our method consistently improves PGP-solving across model scales and achieves strong performance against both general-purpose MLLMs and specialized geometry solvers. Under test-time scaling, CG-MTE achieves comparable accuracy to high-budget self-consistency while reducing sampling cost by up to 8x. Code and data are publicly available at https://github.com/Jason8Kang/ReTTS-PGPS.