Let
be a point on a circle
, and let
be a point distinct from
on the tangent at
to
. Let
be a point not on
such that the line segment
meets
at two distinct points. Let
be the circle touching
at
and touching
at a point
on the opposite side of
from
. Prove that the circumcentre of triangle
lies on the circumcircle of triangle
.
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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 $ABC$.