Cross-ratio

$(z_1,z_2;z_3,z_4)=\dfrac{(z_1-z_3)(z_2-z_4)}{(z_1-z_4)(z_2-z_3)}$ , unchanged by any Möbius transformation.

The cross-ratio of four distinct points $z_1,z_2,z_3,z_4$ in the extended complex plane is $(z_1,z_2;z_3,z_4)=\dfrac{(z_1-z_3)(z_2-z_4)}{(z_1-z_4)(z_2-z_3)}$ , with the usual convention for a factor involving $\infty$ . Its defining property is invariance -- applying any Möbius transformation $T$ to all four points leaves the cross-ratio unchanged. This invariance is what makes the cross-ratio the standard tool for writing down the one Möbius transformation sending three specified points to three target points.

Examples

Setting $(z,z_2;z_3,z_4)=(w,w_2;w_3,w_4)$ and solving for $w$ gives the Möbius transformation carrying $z_2,z_3,z_4$ to $w_2,w_3,w_4$ .
Four points lie on a common generalized circle exactly when their cross-ratio is a real number.