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 $\mathrm{area} \mathrm{of} ∆DEF=\frac{1}{4} (\mathrm{area} \mathrm{of} ∆ABC)$.

Prove that the straight line joining the midpoints of two sides of a triangle is parallel to the third side.

Prove that the straight line joining the midpoints of the oblique sides of a trapezium is parallel to each of the parallel sides.

If the diagonals $AC$ and $BD$ of a quadrilateral $ABCD$ are perpendicular to one another, prove that area of the quadrilateral $=\frac{1}{2}\times AC\times BD$.

If the medians $AD$ and $BE$ of a triangle $ABC$ intersect at $O$, prove that area of $∆AOB=$ area of quadrilateral $CDOE$.