Shared Vertex (Posted on 2013-09-25)

	Submitted by Bractals
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Consider the vertices of the squares as complex numbers. WOLOG let A = 0, B = 2, C = 2(1+i), and D = 2i. For an arbitrary complex number z ≠ 1, let B' = Bz, C' = Cz, and D' = Dz. The midpoints are then          P = (B + D)/2 = 1+i          Q = (B' + D)/2 = i+z          R = (B' + D')/2 = z+iz          S = (B + D')/2 = 1+iz If P+R = Q+S, then the diagonals of the quadrilateral PQRS bisect each other. If (R - P)i = S - Q, then the diagonals are perpendicular and of equal length. If both, then PQRS is a square.          P+R = (1+i) + (z+iz)                   = (i+z) + (1+iz)                   = Q+S          (R - P)i = [(z+iz) - (1+i)]i                       = iz-z-i+1                       = (1+iz) - (i+z)                       = S - Q QED

	Subject	Author	Date
	Coordinatized Solution	Jer	2013-09-27 10:16:15