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\title{Twitch 129.3}
\subtitle{Evan Chen}
\author{Twitch Solves ISL}
\date{Episode 129}
\maketitle
\section*{Problem}
Does there exist a set $S$ of $4$ circles, no three coaxial,
such that there is exactly $4$ circles tangent to all circles in $S$?
\section*{Video}
\href{https://www.youtube.com/watch?v=hA8yUGtN0ks&list=PLi6h8GM1FA6yHh4gDk_ZYezmncU1EJUmZ}{\texttt{https://youtu.be/hA8yUGtN0ks}}
\newpage
\section*{Solution}
Yes. Here is a construction.
Take a scalene triangle $ABC$, and its nine-point circle.
Then there are exactly four circles tangent to line $AB$, $BC$, $CA$,
and the nine-point circle, namely the incircle and its excircles.
Now invert around any point not on the lines $AB$, $BC$, $CA$,
or the nine-point circle to transform this
into four circles with the desired
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