Question 132
Let $BE$ be an altitude of an acute triangle $ABC$ and let $P$ be the point on side $AB$ such that $AP = AE$. Let $N$ be the point for which $BCEN$ is a parallelogram. The areas of the triangles $AEP$ and $BNP$ are equal. Lines $NE$ and $AB$ intersect at $Q$.
a) Prove that the median of triangle $ABC$ drawn from the vertex $C$ intersects the line segment $PQ$.
b) Prove that in the triangle $ABC$, the bisector of the angle $A$, the altitude drawn from the vertex $B$ and the median drawn from the vertex $C$ meet in one point.
a) Prove that the median of triangle $ABC$ drawn from the vertex $C$ intersects the line segment $PQ$.
b) Prove that in the triangle $ABC$, the bisector of the angle $A$, the altitude drawn from the vertex $B$ and the median drawn from the vertex $C$ meet in one point.