The circle
has centre
, and
is a diameter of
. Let
be a point of
such that
. Let
be the midpoint of the arc
which does not contain
. The line through
parallel to
meets the line
at
. The perpendicular bisector of
meets
at
and at
. Prove that
is the incentre of the triangle
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The circle $S$ has centre $O$, and $BC$ is a diameter of $S$. Let $A$ be a point of $S$ such that $\angle AOB<120{{}^\circ}$. Let $D$ be the midpoint of the arc $AB$ which does not contain $C$. The line through $O$ parallel to $DA$ meets the line $AC$ at $I$. The perpendicular bisector of $OA$ meets $S$ at $E$ and at $F$. Prove that $I$ is the incentre of the triangle $CEF.$
Školjka