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{\bf Question}

Let $\ell_1$ be the hyperbolic line contained in the Euclidean
line $\{ z\in {\bf C}\: |\: {\rm Re}(z) = 4\}$, let $\ell_2$ be
the hyperbolic line contained in the Euclidean circle with center
$-3$ and radius $3$, and let $p$ be the point $p = 2 + 2i$.
Determine explicitly all the hyperbolic lines through $p$ which
are parallel to both $\ell_1$ and $\ell_2$.


\medskip

{\bf Answer}

Let's take it one line at a time:

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Consider the two lines through $p$ and the two endpoints at
infinity of $\ell_2$:

The line through $p$ and 0 has endpoints at infinity 0,4.

The line through $p$ and -6 has endpoints at infinity
$-6,\frac{5}{2}$ (of center and radius of the euclidean circle
containing the hyperbolic line through $p$, -6).

These 4 points $-6,0,\frac{5}{2},4$ break $\bf{R} \cup \{\infty\}$
into 4 intervals:
\begin{center}
$(-6,0),\ \left[0,\frac{5}{2}\right],\
\left(\frac{5}{2},4\right),\ [4,-6]$
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(where $[4,-6]=[4,\infty]\cup\{\infty\}\cup(-\infty,-6]$ is an
interval through $\infty$).

A hyperbolic circle through $p$ intersects $\ell_2$ if and only if
it has an end point at infinity in (-6,0) or
$\left(\frac{5}{2},4\right)$.  So, the hyperbolic lines through
$p$ and parallel to $\ell_2$ correspond to points in
$[4,-6]\cup\left[0,\frac{5}{2}\right]$.

Similarly, the two hyperbolic lines through $p$ and the endpoints
at infinity of $\ell_1$ determine 4 intervals on $\bf{R} \cup
\{\infty\}$ namely

$$(0,2),\ [2,4],\ (4,\infty),\ \rm{and}\ [-\infty,0]$$

The lines parallel to $\ell_1$ correspond to the points in $[2,4]
\cup [-\infty,0]$.

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So, the lines through $p$ parallel to both $\ell_1$ and $\ell_2$
correspond to the points in the intersection

$\left([2,4]\cup[-\infty,0]\right)\cap\left([4,-6]\cup[0,\frac{5}{2}]\right)=
[2,\frac{5}{2}]\cup[-\infty,-6]\cup\{0,4\}$

(where $[-\infty,-6]=(-\infty,-6]\cup\{\infty\}$).

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