Define the cross-ratio of four points in , with distinct.
Let be three distinct points. Show that, for every value , there is a unique point with . Let be the set of points for which the cross-ratio is in . Show that is either a circle or else a straight line together with .
A map satisfies
for each value of . Show that this gives a well-defined map with equal to the identity.
When the three points all lie on the real line, show that must be the conjugation map . Deduce from this that, for any three distinct points , the map depends only on the circle (or straight line) through and not on their particular values.