Prove that the line segment joining the midpoints of a pair of opposite sides of a parallelogram divides it into two parallelograms of equal area.

$AD$ 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.

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.