TongGeometry: Automated Geometry Engine

• TongGeometry is a Euclidean geometry system that systematically generates and proves olympiad-level theorems using a guided tree search approach.
• It integrates fine-tuned LLM guidance with DSL encoding to efficiently represent and validate complex geometric constructions.
• The system achieves high performance on IMO benchmarks using consumer-grade hardware while discovering billions of novel theorems.

TongGeometry is a Euclidean geometry system designed for guided tree-search-based problem proposing and automated theorem proving in the domain of mathematics olympiads. It establishes a large-scale engine for both generating novel theorems and discovering valid proof traces with a focus on complex olympiad-level problems, especially those requiring sophisticated auxiliary constructions. TongGeometry's architecture and workflow enable it not only to surpass prior state-of-the-art systems in solving capabilities but also to function as a prolific generator and validator of conjectures and proofs, thereby positioning itself as an effective geometry "coach" rather than merely a solver (Zhang et al., 14 Dec 2024).

1. System Architecture and Design Principles

TongGeometry is constructed to efficiently support both the proposal and automated proof of challenging plane geometry theorems. At its core, the system implements a tree-search paradigm capable of traversing the exponentially large search space of geometric constructions and predicates. It integrates fine-tuned LLMs for guidance, which help prioritize plausible construction and inference steps during proof search.

Key design considerations include minimizing computational overhead per search node, optimizing the representation and storage of geometric configurations, and accommodating a DSL (domain-specific language) closely related to that of AlphaGeometry. All internal geometric objects—points, lines, circles, and predicates—are represented as explicit DSL commands, permitting unambiguous translation between construction sequences, logical inference chains, and eventual proof outputs. No natural-language processing pipeline is included for problem statements; all inputs/outputs are strictly in DSL (Zhang et al., 14 Dec 2024).

2. Theorem Discovery and Guided Tree Search

At the center of TongGeometry's discovery engine is a systematic guided tree search, capable of proposing and validating an extremely large volume of geometry theorems. The system discovered 6.7 billion geometry theorems requiring auxiliary constructions, including 4.1 billion that exhibit geometric symmetry, all within computational budgets comparable to earlier systems (Zhang et al., 14 Dec 2024).

Guidance arises from fine-tuned LLMs that inform the expansion of the search tree, especially in selecting promising auxiliary constructions. The search process is both generative (theorem proposal) and discriminative (proof discovery), inherently allowing the system to filter, validate, and organize vast numbers of potential theorems and proofs. The tree-search mechanism ensures coverage of nontrivial olympiad-style statements that classical brute-force or database-driven approaches find intractable.

3. Problem Representation and DSL Encoding

TongGeometry operates entirely on problems encoded in a domain-specific language. Each problem is formulated as a deterministic sequence of primitive geometric constructions (e.g., “Point A”, “Line AB”, “Circle with center C through D”) followed by a goal predicate (e.g., “eqangle(KIL, XPY)”). Problem statements, auxiliary objects, and proof goals are all explicitly enumerated in DSL; diagrams are neither visualized nor stored (Zhang et al., 14 Dec 2024).

The same DSL structure is used for both input specification and output proof traces. Inputs describe free points, given figures, and the target conclusion; outputs consist of interpreted construction and inference steps leading to proof of the goal predicate. All auxiliary constructions, if required for the proof, are fully specified as additional DSL commands within the proof trace. No additional metadata, topic labels, hints, or lemma annotations are present.

4. Benchmarking and Performance: IMO-AG-30

TongGeometry’s performance is benchmarked against the IMO-AG-30 dataset, which consists of 30 geometry problems spanning 23 years of IMO Short and Long List problems, each manually translated to DSL (Zhang et al., 14 Dec 2024, Sinha et al., 9 Apr 2024). The only complexity proxy provided for IMO-AG-30 is proof trace length; no natural-language versions, topic tags, or difficulty labels are supplied. Each problem is evaluated purely as a sequence of DSL instructions, with no graphical annotations.

TongGeometry’s solve rates on IMO-AG-30 surpass all previous symbolic, neuro-symbolic, and human baselines. It solved all 30 problems (100%) under a wall-clock limit of up to 38 minutes per problem (hardware: 32 CPU cores and 1 RTX 4090 GPU), exceeding the prior state-of-the-art, including AlphaGeometry and all IMO gold medalist averages.

Method	Problems Solved (out of 30)	Hardware
GPT-4o	0	Not specified
AlphaGeometry	25	246 CPU cores + 4 V100 GPUs
TongGeometry (full)	30	32 CPU cores + 1 RTX 4090 GPU
IMO Gold Medalist Avg	25.9	Human

A plausible implication is that TongGeometry’s emphasis on guided auxiliary constructions and its efficient proof search protocol are decisive factors in its comprehensive coverage of IMO-AG-30, outperforming both human top performers and prior AI systems (Zhang et al., 14 Dec 2024).

5. Proposing Problems and Theorem Generation

TongGeometry extends beyond mere problem-solving by systematically proposing previously unknown olympiad‐level geometry theorems. Within the benchmarked computation budget, TongGeometry discovered 6.7 billion geometry theorems, of which 10 were formally proposed to various regional olympiads. Notably, 3 of these proposed theorems were selected for use in real competitions, including national team qualifying exams and prominent regional olympiads in China and the US (Zhang et al., 14 Dec 2024).

Discovery is performed through an automated generative process embedded within the guided tree search, targeting statements that require nontrivial auxiliary configurations and proofs verifiable by the system’s DSL-guided inference engine.

6. Accessibility and Computational Efficiencies

TongGeometry's architecture allows the entire theorem proposing and solving pipeline to run on widely accessible consumer-grade hardware. The complete system operates on a 32-core CPU with a single high-performance GPU (RTX 4090), achieving maximal per-problem times of 38 minutes for even the most challenging tasks in IMO-AG-30 (Zhang et al., 14 Dec 2024).

This computational efficiency marks a substantial reduction in resource requirements compared to AlphaGeometry, which necessitated 246 CPU cores and 4 V100 GPUs to approach comparable coverage. The accessibility of TongGeometry fosters the democratization of advanced geometry proving and discovery, opening the domain to a broader array of users and research environments.

7. Comparative Evaluation and System Significance

Systematic evaluation places TongGeometry at the forefront of automated olympiad geometry theorem discovery and proving. It substantially exceeds the capabilities of prior purely symbolic (e.g., Wu’s method, DD, DD+AR) and state-of-the-art neuro-symbolic systems (AlphaGeometry), not only in proof coverage but also in resource efficiency and the scope of novel theorem generation (Zhang et al., 14 Dec 2024, Sinha et al., 9 Apr 2024).

TongGeometry's duality as both a proving and proposing system, combined with end-to-end automation and minimized hardware demands, establishes a new paradigm in the field of automated geometry. A plausible implication is that future research and benchmarking may increasingly emphasize generative capabilities and scalability alongside competitive solving metrics. No controversies or reported limitations (beyond what is stated in the data) are identified in the referenced works.