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Let the quadrilateral be ABCD.  Diagonals AC and BD iare equal and they ntersect (bisect) at right angles at O.

So  AO = BO = OC = OD = AC/2 = BD/2

In ΔAOB and ΔBOC,  angle AOB = 90 deg = BOC.   AO = BO = CO
Hence both triangles are congruent.  So  AB = BC

Similarly in ΔBOC and Δ COD,  they are congruent and BC = CD
Similarly in  ΔCOD and Δ DOA,  they are congruent and CD = DA

In Δ AOB,   angle OAB = OBA as the sides OA and OB are equal.
       angle OAB = angle OBA = (180 - AOB)/2 = 90/2  = 45 deg

Similarly in ΔBOC, ΔCOD and ΔDOA, we get
     angle OBC = OCB = OCD = ODC = ODA = OAD = 45 deg.

Hence in quadrilateral ABCD, angle A = angle OAD + OAB = 45+45=90 deg.

Angle B = angle OBA + OBC = 45+45 = 90 deg.
Similarly other angles C = D = 90 deg.

Hence ABCD is a square.

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