Question 7
Let $\omega$ be the incircle of a triangle $ABC$ with $\omega$ tangent to $BC$ and $AC$ at the points $D$ and $E$, respectively. Draw a line perpendicular to $BC$ at $D$ to meet the circle $\omega$ at a point $P$ closer to $A$. The line $AP$ meets $BC$ at $M$. Let $N$ be a point on the segment $AC$ such that $AE = CN$, whereas
$BN$ intersects the circle $\omega$ at a point $Q$ closer to $B$ and intersects $AM$ at the point $R$. Show that the area of the triangle $ABR$ is equal to the area of the quadrilateral $PQMN$.
$BN$ intersects the circle $\omega$ at a point $Q$ closer to $B$ and intersects $AM$ at the point $R$. Show that the area of the triangle $ABR$ is equal to the area of the quadrilateral $PQMN$.