= Conjugation of a Möbius transformation with two distinct fixed points
{title2=$\frac{f(z)-\alpha}{f(z)-\beta}=k\frac{z-\alpha}{z-\beta}$}
For $f(z)=(az+b)/(cz+d)$ with distinct finite <fixed points of a Möbius transformation> $\alpha,\beta$, the fixed-point equation gives $f(z)-\alpha=(a-c\alpha)(z-\alpha)/(cz+d)$. Dividing the two identities gives $k=(a-c\alpha)/(a-c\beta)$, which is nonzero because $(a-c\alpha)(a-c\beta)=ad-bc\ne0$. The identities extend over the <Riemann sphere>. This conjugates the transformation to a dilation.
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