Chapter 5 - Wu's method for automated geometry theorem proving and discovering

Abstract

This chapter discusses Wu's method for automated geometry theorem proving and discovering. A brief review of automated geometry theorem proving (AGTP) is provided. A short introduction to the general ideas, principles, and algorithms underlying Wu's method for automated theorem proving in elementary geometry is also presented. Variants and improved versions of the method for proving and discovering different classes of theorems are discussed in this chapter. The two approaches to AGTP include the logic approach and the algebraic computation approach. The theorem combined with Wu-Ritt's zero decomposition theorems provides a complete way of analyzing a geometric theorem. The formulation of AGTP can find the missing nondegeneracy conditions. It addresses the nature of the statement: if a statement is proved to be generally false, it cannot be a theorem no matter how many reasonable additional nondegeneracy conditions are added. Nondegeneracy conditions are usually in algebraic form, and it is hard to transform them into geometric form. A constructive geometric statement is said to be irreducible if it has only one nondegenerate component, and the ascending set for this component is irreducible. The theorem proving in geometry over finite fields is also described.