## Synthetic Construction of Harmonic Quadruples

**Interactive Course on Projective Geometry**

**Definition 0.1**
A quadruple of points on the projective line is said to be a

*harmonic quadruple* if its cross-ratio equals

.

**Theorem 0.1**
In the following figure the quadruple

is harmonic.

**Exercise 0.1** (05)
Find in the figure two perspectives such that their composition
interchanges

and

while fixing both

and

.

**Exercise 0.2** (05)
Use that perspectives preserve cross-ratios to deduce the theorem
from the previous exercise.