A Method for the Automated Discovery of Angle Theorems

Abstract

One approach to Geometric Discovery starts with a given geometry diagram, and hunts systematically, or unsystematically for provable statements about the geometric entities, or further derived geometric entities [1]. The given diagram can be in fact a parametrized family of diagrams [5]. Another approach [4] is to start with the statements one wants to prove, and discover supplementary conditions required to make the theorems true. Again, however, the geometric milieu is given. A problem for such systems is to determine the interestingness of generated theorems, metrics for which are an active topic of research[2]. In this paper, we consider working in reverse and generating the geometric diagram to match a more abstract form of the theorem, which guarantees both its solution, but also a certain level of interestingness. The abstract form is developed by analogy with known theorems, considered (by this author) to be aesthetically pleasing. We develop and automate here a method for generating many theorems of comparable structure but different geometry to our seed theorems. Hopefully this might lend us some control of the richness and tractability, even aesthetic appeal of our generated theorems. Having promised emergent geometry, we immediately limit the scope of our work, however, to consider theorems in the Naive Angle Method employed by Geometry Expressions [9] for angle specific problems. While the method accommodates a number of different constraint types, in the bulk of this paper, we focus solely on the angle bisector constraint, which can be disguised as an isosceles triangle, a circle chord, or a reflection. In any case, it contributes a row with 3 values -1,-1,2 to the constraint matrix. At one level, we can re-interpret the same matrix using different geometry: for example changing a circle chord into an angle bisector (figure 1). At another level, we consider matrices with non zero elements in the same places, but with different assignments within the row of the numerical values (they will still be -1, -1 , 2, only their order will be different). At a third level, we generalize to consider matrices with a similar pattern of non-zero positions. For a class of such matrices, we give structural conditions which determine the presence or absence of theorems of comparable interest to the prototype.