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\begin{document}

{\bf Question}

Determine whether or not the four points $1,\ -3,\ -1,\ -2i,$ and
$3i$ lie on a circle in the Riemann sphere $\overline{C}$.

\medskip

{\bf Answer}

One approach is to use the cross ratio. Another is to proceed
directly, using that M\"{o}bius transformations take circles to
circles. Specifically, construct the M\"{o}bius transformation
taking 1 to , $-3$ to $\infty$, $3i$ to 0, and see what this
transformation does to $-1-2i$; these 4 points lie on a circle if
and only if $-1-2i$ goes to a point on ${\bf{R}}$.

$m(-3)=\infty$ and $m(3i)=0$ give $m(z)=\ds\frac{z-3i}{z+3}$

$m(1)=1$ gives $m(z)=\ds\frac{z-3i}{z+3} \cdot \ds\frac{4}{1-3i}$

Then, \begin{eqnarray*} m(-1-2i) & = & \ds\frac{-1-2i-3i}{-1-2i+3}
\cdot \ds\frac{4}{1-3i}\\ & = & \ds\frac{-4-20i}{(2-2i)(1-3i)}\\ &
= & \ds\frac{-4-20i}{-4-8i} \cdot \ds\frac{-4+8i}{-4+8i}\\ & = &
\ds\frac{16+160+80i-32i}{80} \end{eqnarray*} which is not real.

So, these points \un{do not lie on a circle in
$\overline{\bf{C}}$}.
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