Question 112
A quadrilateral is called cyclic when its four vertices lie on the same circle. Consider a cyclic quadrilateral $ABCD$ with perpendicular diagonals $AC$ and $BD$. Moreover, let $O$ be the center of the circle passing through the vertices of the quadrilateral and $P$ the intersection point of the diagonals.
a) From the point $P$ we draw a line $r$ perpendicular to $BC$. The line $r$ meets $BC$ at $X$ and $AD$ at $M$. Verify that $M$ is the midpoint of side $AD$.
b) From the point $P$ draw the line $s$ perpendicular to $AD$, meeting $AD$ at the point $Y$ and $BC$ at the point $N$. Verify that the quadrilateral $OMPN$ is a parallelogram.
a) From the point $P$ we draw a line $r$ perpendicular to $BC$. The line $r$ meets $BC$ at $X$ and $AD$ at $M$. Verify that $M$ is the midpoint of side $AD$.
b) From the point $P$ draw the line $s$ perpendicular to $AD$, meeting $AD$ at the point $Y$ and $BC$ at the point $N$. Verify that the quadrilateral $OMPN$ is a parallelogram.