Let
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
be a point on a circle
![S_1](/media/m/7/7/e/77ed808eaa71be903a10ce754f90a904.png)
, and let
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
be a point distinct from
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
on the tangent at
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
to
![S_1](/media/m/7/7/e/77ed808eaa71be903a10ce754f90a904.png)
. Let
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
be a point not on
![S_1](/media/m/7/7/e/77ed808eaa71be903a10ce754f90a904.png)
such that the line segment
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
meets
![S_1](/media/m/7/7/e/77ed808eaa71be903a10ce754f90a904.png)
at two distinct points. Let
![S_2](/media/m/c/1/1/c11855875777bfedb764b27ccc108413.png)
be the circle touching
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
at
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
and touching
![S_1](/media/m/7/7/e/77ed808eaa71be903a10ce754f90a904.png)
at a point
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
on the opposite side of
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
from
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
. Prove that the circumcentre of triangle
![BCD](/media/m/3/e/e/3eefa3e34f78e628cbb5cd3988774661.png)
lies on the circumcircle of triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
.
%V0
Let $B$ be a point on a circle $S_1$, and let $A$ be a point distinct from $B$ on the tangent at $B$ to $S_1$. Let $C$ be a point not on $S_1$ such that the line segment $AC$ meets $S_1$ at two distinct points. Let $S_2$ be the circle touching $AC$ at $C$ and touching $S_1$ at a point $D$ on the opposite side of $AC$ from $B$. Prove that the circumcentre of triangle $BCD$ lies on the circumcircle of triangle $ABC$.