Prove that if the lines on which the diagonals of a quadrilateral lie are its axes of symmetry

Prove that if the lines on which the diagonals of a quadrilateral lie are its axes of symmetry, then the quadrilateral is a rhombus.

Let the quadrangle ABCD AC and BD be its diagonals.

Since, by condition, BD is the axis of symmetry, then OA = CO, AB = AD, BC = CD, AC is also the axis of symmetry of the quadrilateral, then BO = BD, AD = CD, AB = BC.

In the quadrangle ABCD, the lengths of all sides are equal, and its diagonals, at the point of intersection, are divided in half, then ABCD is a parallelogram in which all sides are equal, therefore the quadrilateral is a rhombus, which was required to prove.