KöMaL 2014 B4613

The rhombus $A_1 B_1 C_1 D_1$ lies in the interior of a parallelogram ABCD . The vectors $\overrightarrow{AB}$ and $\overrightarrow{A_1B_1}$ have the same direction, and the vectors $\overrightarrow{BC}$ and $\overrightarrow{B_1C_1}$ also have the same direction. Show that ABCD is a rhombus if and only if the sum of the areas of quadrilaterals $AA_1 D_1 D$ and BCC $_1 D_1$ is equal to the sum of the areas of quadrilaterals ABB $_1 A_1$ and CDD $_1 C_1$.