Prove that the diagonals of a parallelogram divide it into four triangle of equal are.

In the figure given below, $AD$ is the median of triangle  $ABC$ and $P$ is any point on $AD$. Prove that the area of the triangle $PBD=$ the area of triangle $PDC$.

In the figure given below, $AD$ is the median of triangle  $ABC$ and $P$ is any point on $AD$. Prove that the area of the triangle $ABP=$ the area of triangle $APC$.

In the figure given below, point $D$ divides the side $BC$ of the triangle $ABC$ in the ratio $m:n$. Prove that $\mathrm{ar}(\mathrm{triangle} ABD):\mathrm{ar}(\mathrm{triangle} ADC)=m:n$

In the figure given below, $\mathrm{ABC}$ is right angled triangle at $A$. $AGFB$ is a square on the side $AB$ and $BCDE$ is a square on the hypotenuse $BC$. If $AN\perp ED$, prove that area of square $\mathrm{ABFG}=$ area of rectangle $BENM$.

$ABCD$is a square. $E$ and $F$ are respectively the midpoints of $\mathrm{BC}$ and $CD$. If $R$ is the mid-point of $EF$ as shown in the figure, prove that $$$\mathrm{ar} \left(\Delta AER\right)=ar \left(AFR\right)$.