AlphaGeometry
In plain wordsA Google DeepMind system from 2024 that solves olympiad geometry problems nearly as well as top human competitors, by pairing a language model that suggests constructions with a logic engine that checks every step.
A neuro-symbolicneuro-symbolic AI, n.: AI that combines neural networks, which learn patterns from data, with symbolic methods, which follow explicit rules and logic, aiming for systems that both learn flexibly and reason reliably. theorem prover for Euclidean plane geometry from Google DeepMindGoogle DeepMind, prop. n.: Google's main artificial intelligence laboratory, formed in 2023 by joining DeepMind and Google Brain. It builds the Gemini models and scientific AI systems such as AlphaFold.. Its symbolic engine, which pairs geometric inference rules with algebra on angles, ratios and distances, works out everything that follows from a diagram; whenever that runs dry, a language modellanguage model, n.: A system that has learned how likely different sequences of words are, so it can predict the next word in a text and, by doing that repeatedly, write text of its own. suggests an auxiliary construction — a new point, line or circle — and deduction starts again. The model learned only from 100 million synthetic theorems and proofs, made by sampling random diagrams and deducing their consequences; no human-written proofs were used. Given 30 olympiad geometry problems from 2000 to 2022, it solved 25 within the time limits, where the best earlier method managed 10 and an average gold medalist about 25.9. AlphaGeometry 2, used at the 2024 IMO and described in a 2025 preprint, extended the formal language, used a GeminiGemini, prop. n.: Google's family of AI models and its chatbot assistant, launched in December 2023. Gemini models handle text, images, audio and video and power many Google products.-based model and solved 84 percent of IMO geometry problems from 2000 to 2024, including the 2024 problem.
AlphaGeometry's auxiliary point looked arbitrary until the symbolic engine closed the proof.
Origin
Described in 'Solving olympiad geometry without human demonstrations' by Trieu Trinh, Yuhuai Wu, Quoc Le, He He and Thang Luong in Nature on 17 January 2024, with code released openly. The successor was reported in a February 2025 preprint.
Related
- AlphaProofAlphaProof, prop. n.: A Google DeepMind system that learned to write computer-checked mathematical proofs by trial and error. In 2024 it helped an AI reach silver-medal level at the International Mathematical Olympiad for the first time.
- neuro-symbolic AIneuro-symbolic AI, n.: AI that combines neural networks, which learn patterns from data, with symbolic methods, which follow explicit rules and logic, aiming for systems that both learn flexibly and reason reliably.
- automated theorem provingautomated theorem proving, n.: Getting computers to prove mathematical or logical statements by themselves, by searching for a chain of valid steps that leads from basic assumptions to the statement.
- synthetic datasynthetic data, n.: Data created by computer programs or AI models rather than collected from the real world — such as practice math problems written by a model, or simulated street scenes for self-driving cars.
- Google DeepMindGoogle DeepMind, prop. n.: Google's main artificial intelligence laboratory, formed in 2023 by joining DeepMind and Google Brain. It builds the Gemini models and scientific AI systems such as AlphaFold.
Sources
- Trinh, T. H., Wu, Y., Le, Q. V., He, H. & Luong, T. (2024). Solving Olympiad Geometry without Human Demonstrations. Nature 625, 476–482.
Mentioned in
Google DeepMindGoogle DeepMind, prop. n.: Google's main artificial intelligence laboratory, formed in 2023 by joining DeepMind and Google Brain. It builds the Gemini models and scientific AI systems such as AlphaFold., International Mathematical OlympiadInternational Mathematical Olympiad, prop. n.: The world's top math competition for high-school students. Because its problems demand creative written proofs, not just answers, earning a gold medal there became a famous test of AI reasoning, first passed in 2025., neuro-symbolic AIneuro-symbolic AI, n.: AI that combines neural networks, which learn patterns from data, with symbolic methods, which follow explicit rules and logic, aiming for systems that both learn flexibly and reason reliably., AlphaProofAlphaProof, prop. n.: A Google DeepMind system that learned to write computer-checked mathematical proofs by trial and error. In 2024 it helped an AI reach silver-medal level at the International Mathematical Olympiad for the first time.