Theorem Discovery Amongst Cyclic Polygons

In [8], a characterization is made of linear systems involving angle bisection conditions which are not full rank. In such a system, one of the conditions is implied by the remainder, and, if the angle bisections are interpreted geometrically, this dependence may be stated in a number of different ways as a geometry theorem. The characterization leads to a catalog of such linear systems. An approach to theorem discovery is proposed wherein a linear system is initially selected, and then interpreted geometrically as a theorem. In [7], a program is described which applies this approach, constructing a particular geometry theorem corresponding to a randomly selected linear system from the catalog. In order to reduce diagram complexity, the program is biased in favor of constructing cyclic polygons wherever possible. In the case where it is able to construct a cyclic polygon using all the rows of the linear system, the theorem which is produced has the following form. Given a cyclic 2n-gon, where n 1 specified pairs of sides are parallel, then a final specified pair of sides is also parallel. For example, in a cyclic hexagon, with two pairs of opposite sides parallel, the third pair of sides is also parallel.