ABCD is a square. On its sides AB, BC, CD and DA, points M, N, P and K are selected

ABCD is a square. On its sides AB, BC, CD and DA, points M, N, P and K are selected, respectively, so that AM = BN = CP = DK. Prove that the quadrilateral MNPK is a square.

Consider a square ABCD.
By connecting the MNPK points, we get a new quadrilateral.
To prove that a quadrangle is a square, you need to prove that all sides of it are equal or all angles are equal.
Consider triangles MBN, NCP, PKD, MAK – they are the same, since they all have 2 identical sides and one identical angle.
If these squares are equal, then the segments mn = nd = dk = km. A quadrangle with the same sides is a square