Focus of Cord.
Tangents of Circle in P, resp. Q:

\[
   \begin{pmatrix}x\\y\end{pmatrix}
   =
   \underline{e}_P+t~\underline{f}_P,
   \quad
   \begin{pmatrix}x\\y\end{pmatrix}
   =
   \underline{e}_Q+s~\underline{f}_Q
\]

Intersects, if and only if: $\theta_P \neq \theta_Q$,
when:

\[
   t ~\underline{f}_P-s~\underline{f}_Q
   =
   \underline{e}_Q-\underline{e}_P
\]

Multiply with $\underline{e}_P$, $\underline{e}_P \cdot~ \underline{f}_P=0$:

\[
   s~ \underline{f}_Q \cdot~ \underline{e}_P
   =
   (\underline{e}_P-\underline{e}_Q)\cdot~ \underline{e}_P
   \quad \Leftrightarrow \quad
   s=\frac{
      \cos{\theta}-1
   }{
      \sin{\theta}
   }
\]

Multiply with $\underline{e}_Q$, $\underline{e}_Q \cdot~ \underline{f}_Q=0$:

\[
   t~ \underline{f}_P \cdot~ \underline{e}_Q
   =
   (\underline{e}_Q-\underline{e}_P)\cdot~ \underline{e}_Q
   \quad \Leftrightarrow \quad
   t=\frac{
      1-\cos{\theta}
   }{
      \sin{\theta}
   }
   =-s
\]

Point of Intersection:

\[
   \begin{pmatrix}x\\y\end{pmatrix}
   =
   \underline{e}_P
   -
   \frac{
      1-\cos{\theta}
   }{
      \sin{\theta}
   }
   \underline{f}_P
\]

Or:
\[
   \begin{pmatrix}x\\y\end{pmatrix}
   =
   \underline{e}_Q
   -
   \frac{
      1-\cos{\theta}
   }{
      \sin{\theta}
   }
   \underline{f}_Q

\]

Double angle formulas:

\[
   t=-s=
   \frac{
      2 \sin^2{\frac{\theta}{2}}
   }{
      2\sin{\frac{\theta}{2}}\cos{\frac{\theta}{2}}
   }
   =
   \tan{\frac{\theta}{2}}
\]

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