Mobius transformation :
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      10 . Let $\Gamma$ be a circle through the points $z_{2}$, $z_{3}$, $z_{4}$. The points $z$ and $z^{*}$ in MATH are said to be symmetric with              respect to $\Gamma$ if MATH      
             Equivalently, we can say that the points $z$ and $z^{*}$ in MATH are symmetric with respect to $\Gamma$ if every circle              passing through $z$ and $z^{*}$ intersects $\Gamma$ orthogonally. It appears that the definition of symmetric points not              only depends on the circle but also on the points $z_{2}$, $z_{3}$, $z_{4}$, but it is not true. The definition of symmetric              points does not depend on the choice of points $z_{2}$, $z_{3}$, $z_{4}$.

Symmetry Principle: If a Mobius transformation $T$ maps a circle $\Gamma_{1}$ onto the circle $\Gamma_{2}$ then any pair of points symmetric with respect to $\Gamma_{1}$ are mapped by $T$ onto a pair of points symmetric with respect to $\Gamma_{2}$.

Example 1: Find the Mobius transformation that maps $0$, $1$, $2$ to $-1$, $0$, $4$ respectively.
Set $z_{1} = 0$, $z_{2} = 1$, $z_{3} = 2$, $w_{1} = -1$, $w_{2} = 0$, $w_{3} = 4$. To find the Mobius transformation $w = T(z)$ such that $T(z_{i}) = w_{i}$ for $i=1$, $2$, $3$, We use the following formula

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