Show that there is only one linear fractional transformation that maps three given distinct points Z1, Z2, Z3 in the extended Z plane onto three specified distinct points W1, W2, W3 in the extended W plane.
Suggestion: Let T and S be two such linear transformations. Then, after pointing out why S^(-1)[T(Zk)] = Zk (K=1,2,3), use the fact that a composition of two linear fractional transformations is a linear fractional transformation and that every linear fractional transformation has at most two fixed points in the extended plane. Use this to show that S^(-1)[T(Z)]=Z for all z. This show that T(z)=S(z) for all z.