The cross-ratio of four points *A*, *B*, *C* and
*D* on a line *l* in projective space is the ratio of the
ratios of the mixed distances |*AC*|:|*AD*| /
|*BD*|:|*BC*| , or, written in bracket notation,
*([AC][BD]):([AD][BC])*.

In the following construction you can see a very important property
of the cross-ratio: It is invariant under projective transformation,
thus if the four points are the intersections of four concurrent lines
with a fifth projection line, the cross-ratio is constant. Move the line by dragging the points marked
* move*.

This gives rise to the natural definition of cross-ratios of four concurrent lines (by the cross-ratio of any four points obtained by intersecting with a fifth line) and also the cross ratio of four points not necessarily collinear with respect to a center of projection.

About the figure: It has been created with Cinderella. Since Cinderella does
not do algebraic calculations out of the box, the necessary divisions
of the lengths are simulated using
von-Staudt-constructions. Von-Staudt-constructions have been used by
(guess who) von-Staudt to show the field structure of
point-line-construction without refering to coordinates. The points
*|AC|*, *|AD|*,... are at distance *|AC|*,
*|AD|*,... from the point *0*, the two intermediate results
*|AC|:|AD|* and *|BD|:|BC|* are at the appropriate distances
from *0*, and the the fat point labelled with the cross-ratio is
at cross-ratio-distance from *0*. While moving the intersection
line, the cross-ratio-point does not move, which shows that its
position (or distance from *0* is invariant under projective
transformations of the points *A*, *B*, *C* and
*D*.