Let
be an acute-angled triangle with
and let
be the centre of its circumcircle
. Let
be a point on the line segment
such that
. Let
be the second point of intersection of
and the line
. If
and
are the midpoints of the line segments
and
, respectively, show that the points
and
are collinear.
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Let $ABC$ be an acute-angled triangle with $AB<AC$ and let $O$ be the centre of its circumcircle $\omega$. Let $D$ be a point on the line segment $BC$ such that $\angle BAD = \angle CAO$. Let $E$ be the second point of intersection of $\omega$ and the line $AD$. If $M, N$ and $P$ are the midpoints of the line segments $BE, OD$ and $AC$, respectively, show that the points $M, N$ and $P$ are collinear.
Školjka