Trisectors!

Geometry Level 5

Let ABCDABCD be a square. Points E,F,G,HE, F, G, H are on the line segments AB,BC,CD,DAAB, BC, CD, DA respectively such that 2AE=EB2AE = EB, 2BF=FC2BF = FC, 2CG=GD2CG = GD, and 2DH=HA2DH = HA. The area bounded by the line segments AG,BH,CE,DFAG, BH, CE, DF is a square with side length 1. If ABAB can be written as aba \sqrt{b}, where aa and bb are positive integers such that bb is square-free, find a+ba+b.

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