Question 151
A triangle $ABC$ is given. Let $M$ be the midpoint of the side $AC$ of the triangle and $Z$ the image of point $B$ along the line $BM$. The circle with center $M$ and radius $MB$ intersects the lines $BA$ and $BC$ at the points $E$ and $G$ respectively. Let $H$ be the point of intersection of $EG$ with the line $AC$, and $K$ the point of intersection of $HZ$ with the line $EB$. The perpendicular from point $K$ to the line $BH$, intersects the lines $BZ$ and $BH$ at the points $L$ and $N$, respectively.
If $P$ is the second point of intersection of the circumscribed circles of the triangles $KZL$ and $BLN$, prove that the lines $BZ$, $KN$ and $HP$ intersect at a common point.
If $P$ is the second point of intersection of the circumscribed circles of the triangles $KZL$ and $BLN$, prove that the lines $BZ$, $KN$ and $HP$ intersect at a common point.