Mobius transformation :
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      (Recall that a circle in the extended complex plane passing through $\infty$ corresponds to a straight line in $\QTR{Bbb}{C}$.         Hence, the word `circles' include the `straight lines' also).
        Result: Let $\Gamma$ be a circle passing through the points $z_{2}$, $z_{3}$ and $z_{4}$. Then, the point $z_{1}$ is on $\Gamma$ if and only if         the cross ratio MATH is a real number.
        Result: The cross ratio of four points MATH is invariant under any Mobius transformations. That is, MATH       7. A Mobius transformaion maps circles onto circles.

      8 . For any given circles $\Gamma$ and $\Gamma^*$ in MATH, there is a Mobius transformation $T$ such that MATH.            Furhtermore, we can specify that $T$ take any three points on $\Gamma$ onto any three points of $\Gamma^{*}$. If we specify the            points then $T$ is unique.

      9 . If $\Gamma$ is a circle then an orientation for $\Gamma$ is an ordered triple of points MATH such that each $z_{j}$ is in $\Gamma$.           These three points not only determine the circle $\Gamma$ uniquely but also give a direction (without any ambiguity)            to $\Gamma$, by proceeding through the points $z_{1}$, $z_{2}$, $z_{3}$ in succession.
           Orientation Principle: Let $\Gamma_{1}$ and $\Gamma_{2}$ be two circles in MATH and let $T$ be a Mobius transformation such that            MATH. Let MATH be an orientation for $\Gamma_{1}$. Then $T$ maps the right side of $\Gamma_{1}$ onto the right            side of $\Gamma_{2}$ and the the left side of $\Gamma_{1}$ onto the left side of $\Gamma_{2}$ with respect to the orientationMATH
 
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