Point and Circle Configurations; A New Theorem

Abstract

Several of the n3 and configurations-n points, n lines, 3 points on a line, 3 lines through a point-are well known, e.g., the 93 figure that illustrates the theorem of Pappus, and the 103 figure that illustrates the theorem of Desargues. (The self-dual comes from the fact that point and line can be interchanged in the above description without changing the meaning.) More general and configurations, ones that are not self-dual, occur as illustrations of many other projective geometry theorems. Some of these, such as the following, appear to be very complicated, but they give an idea of the type of investigation that interested geometers in the late nineteenth century.