Problem: Two circles $\omega_1$ and $\omega_2$, of equal radius
intersect at different points $X_1$ and $X_2$.
Consider a circle $\omega$ externally tangent to $\omega_1$ at $T_1$
and internally tangent to $\omega_2$ at point $T_2$.
Prove that lines $X_1T_1$ and $X_2T_2$ intersect at a point lying on $\omega$.

Solution: We present two solutions, one by homothety and another by inversion.

\begin{center}
\begin{asy}
/*
Converted from GeoGebra by User:Azjps using Evan's magic cleaner
https://github.com/vEnhance/dotfiles/blob/main/py-scripts/export-ggb-clean-asy.py
*/
size(8cm);
pen zzttqq = rgb(0.6,0.2,0.);
pen dcrutc = rgb(0.86274,0.07843,0.23529);
pen aqaqaq = rgb(0.62745,0.62745,0.62745);
pair X_1 = (0.5,0.62449), X_2 = (0.5,-0.62449), M = (0.5,0.), T_2 = (1.35807,0.71538), T_1 = (0.76813,0.22354);
draw(circle((0.,0.), 0.8), linewidth(0.6));
draw(circle((1.,0.), 0.8), linewidth(0.6));
filldraw(circle((1.17050,0.34064), 0.41906), paleyellow, brown);
draw(X_2--T_2, linewidth(0.6) + linetype("2 2") + dcrutc);
draw(X_1--(0.90858,0.01351), linewidth(0.6) + linetype("2 2") + dcrutc);
draw(X_1--X_2, linewidth(0.6) + aqaqaq);
draw(M--T_2, gray);

dot("$X_1$", X_1, dir(90));
dot("$X_2$", X_2, dir(-90));
dot("$M$", M, dir(180));
dot("$T_2$", T_2, dir((1.076, 2.708)));
dot("$T_1$", T_1, dir(170));
\end{asy}
\end{center}

\paragraph{First solution (homothety).}
Consider the composition of homotheties
\[ \omega_1 \xrightarrow{T_1} \omega \xrightarrow{T_2} \omega_2. \]
This is a negative homothety, and since $\omega_1$ and $\omega_2$ have equal radius,
it follows that the composition is just reflection
about the midpoint of $\ol{X_1X_2}$.
In particular, it sends $X_1$ to $X_2$, and thus $\ol{X_1T_1} \cap \ol{X_2T_2}$
is just the image of $X_1$ under the first homothety.

\paragraph{Second solution (inversion).}
Invert around $X_1$. Then we have the following:
\begin{itemize}
  \ii $\omega_1^\ast$ and $\omega_2^\ast$ are lines intersecting at $X_2^\ast$,
  and $X_1$ is a point on the external angle bisector of the two lines.
  \ii The circle $\omega^\ast$ becomes tangent to the two lines
  (in a region adjacent to $X_1$).
  \ii Let $P = \ol{T_1^\ast X_1} \cap \omega^\ast$.
  Then one wishes to show $X_1 X_2^\ast P T_2^\ast$ are concyclic.
\end{itemize}
To show this, notice that
\[ \dang T_2^\ast X_2^\ast X_1 = \dang X_2^\ast T_2^\ast T_1^\ast
  = \dang T_2^\ast P T_1^\ast = \dang T_2^\ast P X_1. \]