Proof of Möbius transformation cross-ratio preservation theorem

From the definition of Möbius transform we get that the image $w_{k}$ of a point $z_{k}$ is

w_{k}=\frac{az_{k}+b}{cz_{k}+d}

From this we get

w_{i}-w_{j}=\frac{az_{i}+b}{cz_{i}+d}-\frac{az_{j}+b}{cz_{j}+d}=\frac{(ad-bc)(% z_{i}-z_{j})}{(cz_{i}+d)(cz_{j}+d)}

and by inserting this into the cross-ratios

\frac{(w_{1}-w_{2})(w_{3}-w_{4})}{(w_{1}-w_{4})(w_{3}-w_{2})}=\frac{\frac{(ad-% bc)(z_{1}-z_{2})}{(cz_{1}+d)(cz_{2}+d)}\frac{(ad-bc)(z_{3}-z_{4})}{(cz_{3}+d)(% cz_{4}+d)}}{\frac{(ad-bc)(z_{1}-z_{4})}{(cz_{1}+d)(cz_{4}+d)}\frac{(ad-bc)(z_{% 3}-z_{2})}{(cz_{3}+d)(cz_{2}+d)}}=\frac{(z_{1}-z_{2})(z_{3}-z_{4})}{(z_{1}-z_{% 4})(z_{3}-z_{2})}

Title	Proof of Möbius transformation cross-ratio preservation theorem
Canonical name	ProofOfMobiusTransformationCrossratioPreservationTheorem
Date of creation	2013-03-22 14:08:20
Last modified on	2013-03-22 14:08:20
Owner	Johan (1032)
Last modified by	Johan (1032)
Numerical id	5
Author	Johan (1032)
Entry type	Proof
Classification	msc 30E20