  Autoformalization involves automatically translating informal math into
formal theorems and proofs that are machine-verifiable. Euclidean geometry
provides an interesting and controllable domain for studying autoformalization.
In this paper, we introduce a neuro-symbolic framework for autoformalizing
Euclidean geometry, which combines domain knowledge, SMT solvers, and large
language models (LLMs). One challenge in Euclidean geometry is that informal
proofs rely on diagrams, leaving gaps in texts that are hard to formalize. To
address this issue, we use theorem provers to fill in such diagrammatic
information automatically, so that the LLM only needs to autoformalize the
explicit textual steps, making it easier for the model. We also provide
automatic semantic evaluation for autoformalized theorem statements. We
construct LeanEuclid, an autoformalization benchmark consisting of problems
from Euclid's Elements and the UniGeo dataset formalized in the Lean proof
assistant. Experiments with GPT-4 and GPT-4V show the capability and
limitations of state-of-the-art LLMs on autoformalizing geometry problems. The
data and code are available at https://github.com/loganrjmurphy/LeanEuclid.
