Python, Straightforward with Explanation

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    Consider the square with side-length S placed in any orientation and location on the plane. The pairwise distances between points (in sorted order) are: S, S, S, S, S*sqrt(2), S*sqrt(2). With some thinking, we can prove this is a necessary and sufficient condition [details after]. Therefore, we should only check this condition, plus that S > 0. For convenience, we'll work with the squares of distances instead.

    def validSquare(self, p1, p2, p3, p4):
        def dist(P, Q):
            return (P[0] - Q[0])**2 + (P[1] + Q[1])**2
        D = [dist(p1, p2), dist(p1, p3), dist(p1, p4),
             dist(p2, p3), dist(p2, p4), dist(p3, p4)]
        return 0 < D[0] == D[1] == D[2] == D[3] and 2*D[0] == D[4] == D[5]

    Let's now prove our condition was necessary and sufficient.

    Suppose points ABCD have pairwise distances in sorted order S, S, S, S, S*sqrt(2), S*sqrt(2). We want to show ABCD is a square. Let us call S a "small side" and S*sqrt(2) a "large side". Without loss of generality, suppose AC is a large side. If BD is a large side, then AB and BC are small sides, so B lies on the intersection of circles between A and C; similarly, D lies on the same intersection, and thus ABCD is a square (as two different circles of the same radius only intersect in two different points).

    All other cases (AB, AD, CB, CD having a large side) are similar, so we'll only consider AB having a large side. Then, BC is the base of icoceles triangles ABC and DBC, so A and D must lie on the perpendicular bisector to BC. D must be closer to BC than A is (otherwise |DB| > |DA| is a contradiction). Since BDC is an equilateral triangle, angle ∠BDC = 60°, and therefore equal angles ∠ADB + ∠ADC = 300° and ∠ADB = 150°. But the sidelengths of triangle ADB are known, which uniquely determines ∠ADB = 90°, a contradiction.

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