Join the diagonal BD.
From the figure we know that AC and BD are the diagonals intersecting at point O.
We know that the diagonals of a parallelogram bisect each other.
Thus, we get O as the midpoint of AC and BD.
Median of triangle divides it into two triangles having equal area.
Consider △ ABD
We know that OA is the median
So we get
Area of △ AOD = Area of △ AOB ……. (1)
Consider △ BPD
We know that OP is the median
So we get
Area of △ OPD = Area of △ OPB …….. (2)
By adding both the equations we get
Area of △ AOD + Area of △ OPD = Area of △ AOB + Area of △ OPB
So we get
Area of △ ADP = Area of △ ABP
Therefore, it is proved that ar (△ ADP) = ar (△ ABP).