A circle
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
has two parallel tangents
![L'](/media/m/8/0/2/8021a6fdd82a9bc80dcf10a5cbcc63ea.png)
and
![L"](/media/m/f/6/c/f6c6f1479131114e2b8a2e7236814782.png)
. A circle
![C'](/media/m/0/0/1/001d1a1af4c90ceda662e79e88845742.png)
touches
![L'](/media/m/8/0/2/8021a6fdd82a9bc80dcf10a5cbcc63ea.png)
at
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
and
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
at
![X](/media/m/9/2/8/92802f174fc4967315c2d8002c426164.png)
. A circle
![C"](/media/m/9/2/4/9247ea4559e1f0de5960071a8b1a09cc.png)
touches
![L"](/media/m/f/6/c/f6c6f1479131114e2b8a2e7236814782.png)
at
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
,
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
at
![Y](/media/m/3/b/c/3bc24c5af9ce86a9a691643555fc3fd6.png)
and
![C'](/media/m/0/0/1/001d1a1af4c90ceda662e79e88845742.png)
at
![Z](/media/m/7/9/4/794ff2bd637e30ea27e50e57eecd0b76.png)
. The lines
![AY](/media/m/b/b/9/bb90496e08c85f2dceaaf0a186d021fe.png)
and
![BX](/media/m/3/3/7/337e22dcd76e2ed929a1109e8f7c9176.png)
meet at
![Q](/media/m/4/5/c/45ce8d14aa1eb54f755fd8e332280abd.png)
. Show that
![Q](/media/m/4/5/c/45ce8d14aa1eb54f755fd8e332280abd.png)
is the circumcenter of
%V0
A circle $C$ has two parallel tangents $L'$ and$L"$. A circle $C'$ touches $L'$ at $A$ and $C$ at $X$. A circle $C"$ touches $L"$ at $B$, $C$ at $Y$ and $C'$ at $Z$. The lines $AY$ and $BX$ meet at $Q$. Show that $Q$ is the circumcenter of $XYZ$