A quadrilateral $ABCD$ is given, $AD=BC$. The projections of $AD$ and $BC$ on $AC$ are equal. Is $ABCD$ parallelogram?
Let $DD_1\perp AC$ and $BB_1\perp AC$. We have $AD_1=CB_1$. I found that the triangles $\triangle AD_{1}D$ and $\triangle BB_{1}C$ are congruent. Does this help and how?