# CHAPTER 15

# CROSS RATIOS

# 15.1 Cross Ratios

**Directed Distances**

Recall that we denote directed distances, or signed distances, with a bar over the distance, as in , and that

Also recall the following facts:

**1.** Given points

*A, B*, and

*C* on a line, if

=

, then

*B* =

*C.*
**2.** Given points

*A, B, C*, and

*X* on a line, if

/

=

, then

*B* =

*X*.

### Directed Distances and Harmonic Conjugates

Recall that given points *A, B, C*, and *D*, then *B* and *D* are harmonic conjugates with respect to *A* and *C* if and only if

as in the figure below.

Here, we are using unsigned distances; for *signed distances, B* and *D* are harmonic conjugates ...