# 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 ...