ABSTRACT
The material in this chapter and the next will extend the results in Euclidean geometry introduced in Chapter 5, to include a few classical theorems in an area that has come to be known as modern geometry. This theory includes a development of the Euclidean plane which has taken place over approximately the last 300 years. It goes far beyond the selfimposed bounds of the ancient Greek geometers. We have chosen to focus our attention on the two most common geometric objects: the triangle and the circle. The interplay between them is seemingly inexhaustible, with new relationships and varying viewpoints being explored and appearing in the literature year after year. We are going to consider here a few of the more prominent ones that have become famous over time, such as the nine-point circle, the collinearity theorems of Menelaus and Ceva, and systems of orthogonal circles. Properties of directed distance and circles will be considered Šrst, to provide the background needed for the other more advanced concepts.
