All the angles of a quadrilateral are equal. What special name is given to this quadrilateral?

$A$ square is inscribed in an isosceles right triangle so that the square and the triangle have one angle common. Show that the vertex of the square opposite the vertex of the common angle bisects the hypotenuse.

In the figure,$X$ and $Y$ are respectively the mid-points of the opposite sides $AD$ and $BC$ of a parallelogram $ABCD$. Also,$BX$ and $DY$ intersect $AC$ at $P$ and $Q$ respectively. Show that $AP = PQ = QC$.

In a rhombus $ABCD$,if $AB = 13 \text{ cm}$,then the perimeter of $ABCD = \ldots \ldots \ldots \text{ cm}$.

In the figure,$P$ is the mid-point of side $BC$ of a parallelogram $ABCD$ such that $\angle BAP = \angle DAP.$ Prove that $AD = 2 CD.$

Prove that the quadrilateral formed by the bisectors of the angles of a parallelogram is a rectangle.