Classical Geometry: Euclidean, Transformational, Inversive, and Projective

CHAPTER 4

THEOREMS OF CEVA AND MENELAUS

4.1 Directed Distances, Directed Ratios

This chapter provides a more algebraic approach to concurrency and collinearity through the theorems of Ceva and Menelaus. The theorems are best understood using the notions of directed distances and directed ratios. We repeat the definition given in the previous chapter.

Let l be a line and assign a direction to the line For two points A and B on the line, with A ≠ B, let AB be the distance between A and B. The directed distance or signed distance from A to B, denoted , is defined as follows:

The directed distances and for the given direction along l are shown in the figure below, where AB = 2.

Properties of Directed Distance

(1) .

(2) For points A, B, and C on a line, .

(3) If , then B = C.

Sometimes property ...