$\mathrm{ABCD}$ is a parallelogram and $\mathrm{E}$ is any point on $\mathrm{AB}$. If $\mathrm{DE}$ is produced meets $\mathrm{CB}$ produced at $\mathrm{F}$, prove that the area of triangle $\mathrm{AEF}=$ area of triangle$\mathrm{BEC}$.

A trapezium$\mathrm{ABCD}$ has sides $AB$ and $DC$ parallel. A straight line parallel to the diagonal $\mathrm{AC}$ cuts $AB$ at $E$and$F$. Prove that triangles $AED$ and $ACF$are equal in area.

Prove that the parallelogram formed by joining the midpoints of the adjacent sides of a quadrilateral is half of the latter.

In figure, $AB\parallel DC\parallel EF, AD\parallel BC$ and $ED\parallel FA$. Prove that the area of $DEFH$is equal to the area of $ABCD$.

In the given figure,$\mathrm{M}$and $\mathrm{N}$are the midpoints of the sides$\mathrm{DC}$ and $\mathrm{AB}$ of the parallelogram $\mathrm{ABCD}$ and area of the parallelogram $\mathrm{ABCD}$ is $36 {\mathrm{cm}}^2$.

If the area of the triangle $\mathrm{BEC}$ is $k c{m}^2$ then find the value of $k$.

In the given figure,$\mathrm{M}$and $\mathrm{N}$are the midpoints of the sides$\mathrm{DC}$ and $\mathrm{AB}$ of the parallelogram $\mathrm{ABCD}$ and area of the parallelogram $\mathrm{ABCD}$ is $36 {\mathrm{cm}}^2$.

Name the parallelogram which is equal in area to the triangle $\mathrm{BEC}$.