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Let
be a circle with center
. Let
and
be two fixed points on the circle
and let
be a variable point on
. We denote by
the intersection point of lines
and
, assuming
. Let
be the circumcircle of triangle
. Let
be the second intersection point of
with
. The tangent to
at
intersects
at
. The line
intersects
at
. The perpendicular bisector of segment
intersects line
at
, and line
intersects
at
. We denote by
the second intersection point of the circumcircle of triangle
with
. Prove that, as point
varies, points
and
remain fixed.


































