How can you prove that the midpoint of the square will be Origin?Vertices A,B,C,D and the midpoints of the sides of the square ABCD lie on x2y2=1.No other points of the square lies on the curve.How can you prove that the midpoint of the square will be Origin ?My approach: If I can show if (x,y) are the coordinates of A then (y, -x) will be so of B , I will be done. I supposed A lies on first quadrant , B lies on second quadrant etc..Can anyone tell please me how to show it ?

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zalutaloj9a0f · Accepted Answer

You can try to argue geometrically using symmetries or you can try to find the coordinates of the vertices of the square.Let (x,y) be a vertex of the square in the first quadrant with smallest x. Then consider the point (y,-x). This point lies on the curve (in the second quadrant) and asking that the midpont (x+y2,y−x2) lies on the curve yields the equation x4+y4=18. Substitution of y2=1x2 yield as solution (take the positive x with the smallest value) that x=5−2 and y=5−2−1. Now you can check that the vertices (x,y),(y,−x),(−x,−y) and (−y,x) form a square satisfying the requirements, and its midpoint is the origin.

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