Automatically Building Diagrams for Olympiad Geometry Problems

TL;DR

This paper introduces GMB, a novel system that automatically constructs diagrams for olympiad geometry problems using a new language and optimization techniques, achieving high success on past IMO problems.

Contribution

The paper presents GMB and GMBL, enabling automated diagram generation for complex olympiad geometry problems with a new domain-specific language and optimization approach.

Findings

Successfully generated diagrams for 94% of 39 IMO problems since 2000.

First automated system capable of modeling complex olympiad geometry problems.

Achieved reliable diagram construction using gradient descent optimization.

Abstract

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.