Let
be a triangle with
and circumcenter
. The bisector of
intersects
at
. Let
be the reflection of
with respect to the midpoint of
. The lines through
and
perpendicular to
intersect the lines
and
at
and
respectively. Prove that the quadrilateral
is cyclic.
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Let $ABC$ be a triangle with $AB \neq AC$ and circumcenter $O$. The bisector of $\angle BAC$ intersects $BC$ at $D$. Let $E$ be the reflection of $D$ with respect to the midpoint of $BC$. The lines through $D$ and $E$ perpendicular to $BC$ intersect the lines $AO$ and $AD$ at $X$ and $Y$ respectively. Prove that the quadrilateral $BXCY$ is cyclic.
Školjka