$ABCD$ and $ABPQ$ are parallelogram such that the points $C, D, P, Q$ are collinear and $R$ is any point on the side BP. Show that area of $△ARQ=\frac{1}{2}$area of parallelogram $ABCD$.

$AD$ is a median of a triangle $ABC$ and $P$ is any point on $AD$. Show that $\mathrm{area} \mathrm{of} ∆ABP=\mathrm{area} \mathrm{of} ∆ACP$

$ABC$ is a triangle and $DE$ is drawn parallel to $BC$, cutting the other sides at $D$ and $E$. Join $BE$ and $CD$. Prove that $\mathrm{area}\left(∆DBC\right)=\mathrm{area}\left(∆EBC\right)$.

$ABC$ is a triangle and $DE$ is drawn parallel to $BC$, cutting the other sides at $D$ and $E$. Join $BE$ and $CD$. Prove that $\mathrm{area}\left(∆BDE\right)=\mathrm{area}\left(∆CDE\right)$.

Show that the diagonals of a parallelogram divide it into four triangles of equal area.

Show that the area of a rhombus is half the product of the lengths of its diagonals.

Prove that the area of a trapezium is half the product of the sum of the lengths of the parallel sides and distance between them.

The diagonals $AC$ and $BD$ of a quadrilateral $ABCD$ intersect at $O$. If $BO=OD$, prove that $\mathrm{area} \mathrm{of} ∆ABC=\mathrm{area} \mathrm{of} ∆ADC$.

$D, E, F$ are the midpoints of the sides $BC, CA, AB$ respectively of a triangle $ABC$. Prove that $BDEF$ is a parallelogram whose area is half that of $∆ABC$ and $\mathrm{area} \mathrm{of} ∆DEF=\frac{1}{4} (\mathrm{area} \mathrm{of} ∆ABC)$.