Proof:

Let ABCD be a quadrilateral with diagonals AC and BD. We are given that the diagonals are equal in length and bisect each other at right angles.

Step 1: Prove that AB = BC = CD = DA
Since the diagonals AC and BD bisect each other, we can consider the triangles ABD and BCD.

In triangle ABD, the diagonals AC and BD bisect each other at point O. So, AO = BO and DO = CO.

Similarly, in triangle BCD, the diagonals AC and BD bisect each other at point O. So, BO = CO and DO = AO.

Combining these equalities, we have AO = BO = CO = DO.

Thus, all sides of the quadrilateral ABCD are equal in length, i.e., AB = BC = CD = DA.

Step 2: Prove that ABCD is a parallelogram
Since AB = CD and BC = DA, we can conclude that opposite sides of the quadrilateral ABCD are equal in length.

Now, consider the triangles ABC and CDA. In triangle ABC, AB = BC and angle B = angle C (due to right angles formed by the diagonals).

Similarly, in triangle CDA, CD = DA and angle C = angle D.

Combining these equalities, we have AB = BC = CD = DA and angle B = angle C = angle D.

Thus, opposite sides and opposite angles of the quadrilateral ABCD are equal, making it a parallelogram.

Step 3: Prove that ABCD is a rectangle
Since ABCD is a parallelogram, opposite sides are parallel.

Consider the diagonals AC and BD. We are given that they bisect each other at right angles.

If opposite sides of a quadrilateral are parallel and the diagonals bisect each other at right angles, then the quadrilateral is a rectangle.

Thus, ABCD is a rectangle.

Step 4: Prove that ABCD is a square
Since ABCD is a rectangle, all angles are right angles.

We have already established that AB = BC = CD = DA.

If all sides of a quadrilateral are equal in length and all angles are right angles, then the quadrilateral is a square.

Hence, ABCD is a square.

Conclusion:
We have shown that if the diagonals of a quadrilateral are equal in length and bisect each other at right angles, then the quadrilateral is a square.