Automated Theorem Proving in the Homogeneous Model with Clifford Bracket Algebra

A Clifford algebra has three major multiplications: inner product, outer product and geometric product. Accordingly, the same Clifford algebra has three versions: Clifford vector algebra, which features on inner products and outer products of multivectors; Clifford bracket algebra, which features on pseudoscalars and inner products of vectors; Clifford geometric algebra, which features on geometric products of vectors and multivectors. Since 1993, there have been various applications of Clifford vector algebras in automated geometric theorem proving [9], [10], [11], [12], [13], [14], [3], [15], [16], [17], [18]. Some famous theorems which are difficult to prove mechanically by other methods, either algebraic or logical ones, have been proved successfully with Clifford vector algebra methods. 2D Clifford geometric algebra is just the algebra of complex numbers, its applications in automated theorem proving can be found in [2], [9], etc. For projective geometry, bracket algebra is very important for invariant geometric computing [24], [25], [26],[27] and automated theorem proving [21], [22]. For Euclidean geometry, complex bracket algebra [21], distance algebra [4] and the method of areas and Pythagorean distances [2], have been efficiently used in theorem proving. From the brief summary, we notice that Clifford bracket algebra, whose foundation is a set of generalized Grassmann-Plücker relations, has not been systematically applied to theorem proving in Euclidean geometry. For Euclidean geometry, a very useful Clifford algebraic model is the homogeneous model [5], [20], [19]. The Clifford bracket algebra in the homogeneous model is what we choose for invariant and automated theorem proving in Euclidean geometry. This chapter presents some of our newest results in applying this Clifford bracket algebra in 2D Euclidean geometric theorem proving. The highlight is that some tremendously difficult geometric computing tasks can be finished with the Clifford bracket algebra, but not by any other pure algebraic methods when running on currently available computer systems. The following is a typical example – the five-circle theorem: Let A,B, C, D, E be five generic points in the plane. Let