Automatic theorem generation in plane geometry

Abstract

We introduce a conceptual framework for discovery of theorems in geometry and a mechanism which systematically discovers such theorems. Our mechanism incrementally generates geometrical situations, makes conjectures about them, uses a geometry theorem prover to determine the consistency of situations, and keeps valid conjectures as theorems. We define geometry situations, situation descriptions, theorems, and their relationships important to understand our discovery task. An exhaustive generator of situation descriptions has enormous combinatorial complexity. We analyze various ways to reduce that complexity. Ideally, the generator should create a single description of each situation, should generate more general situations before more specific ones, and should use the previously discovered theorems to constrain its generation mechanism. We describe our generator which possesses most of these properties, and we outline further improvements. Our theorem prover is based on Wu's algebraic method for proving geometry theorems. We discuss the interface between our situation generator and theorem prover and the limitations of our discovery system. Examples of theorems discovered by our system are also presented.