Problem #49

A spiral path is constructed in the unit square whose vertices have coordinates (0,0), (1,0), (1,1), and (0,1) as follows. Let A1 = (0,1), A2 = (1,1), A3 = (1,0), and A4 = (0,0). Let A5 be the midpoint of A1A2, A6 be the midpoint of A2A3, A7 be the midpoint of A3A4, etc. This forms a spiral polygonal path A1A2A3A4... converging to a unique point in the plane. Find the coordinates of this point.

Source: Len Bos and Bill Sands

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