# Automatically Building Diagrams for Olympiad Geometry Problems

arXiv: https://arxiv.org/abs/2012.02590 · HTML: https://mlsanity.com/paper/2012.02590/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Dec 2020.

## Built on by (most cited first, 8 of 8)

- [Gold-medalist Performance in Solving Olympiad Geometry with AlphaGeometry2](https://mlsanity.com/paper/2502.03544/) (2025)
- [Diagram Formalization Enhanced Multi-Modal Geometry Problem Solver](https://mlsanity.com/paper/2409.04214/) (2024)
- [NeSyGeo: A Neuro-Symbolic Framework for Multimodal Geometric Reasoning Data Generation](https://mlsanity.com/paper/2505.17121/) (2025)
- [A Survey of Deep Learning for Geometry Problem Solving](https://mlsanity.com/paper/2507.11936/) (2025)
- [GeoFM: Enhancing Geometric Reasoning of MLLMs via Synthetic Data Generation through Formal Language](https://mlsanity.com/paper/2510.27448/) (2025)
- [GeoBuildBench: A Benchmark for Interactive and Executable Geometry Construction from Natural Language](https://mlsanity.com/paper/2605.13167/) (2026)
- [Enhancing Geometric Perception in VLMs via Translator-Guided Reinforcement Learning](https://mlsanity.com/paper/2602.22703/) (2026)
- [Geometric Reasoning in the Embedding Space](https://mlsanity.com/paper/2504.02018/) (2025)

## Abstract (arXiv)

We present a method for automatically building diagrams for olympiad-level geometry problems and implement our approach in a new open-source software tool, the Geometry Model Builder (GMB). Central to our method is a new domain-specific language, the Geometry Model-Building Language (GMBL), for specifying geometry problems along with additional metadata useful for building diagrams. A GMBL program specifies (1) how to parameterize geometric objects (or sets of geometric objects) and initialize these parameterized quantities, (2) which quantities to compute directly from other quantities, and (3) additional constraints to accumulate into a (differentiable) loss function. A GMBL program induces a (usually) tractable numerical optimization problem whose solutions correspond to diagrams of the original problem statement, and that we can solve reliably using gradient descent. Of the 39 geometry problems since 2000 appearing in the International Mathematical Olympiad, 36 can be expressed in our logic and our system can produce diagrams for 94% of them on average. To the best of our knowledge, our method is the first in automated geometry diagram construction to generate models for such complex problems.