Problem
2020 AMC 10B Problem 21
In square ABCD, points E and H lie on \overline{AB} and \overline{DA}, respectively, so that AE=AH. Points F and G lie on \overline{BC} and \overline{CD}, respectively, and points I and J lie on \overline{EH} so that \overline{FI} \perp \overline{EH} and \overline{GJ} \perp \overline{EH}. See the figure below. Triangle AEH, quadrilateral BFIE, quadrilateral DHJG, and pentagon FCGJI each has area 1. What is FI^2? ![[asy] real x=2sqrt(2); real y=2sqrt(16-8sqrt(2))-4+2sqrt(2); real z=2sqrt(8-4sqrt(2)); pair A, B, C, D, E, F, G, H, I, J; A = (0,0); B = (4,0); C = (4,4); D = (0,4); E = (x,0); F = (4,y); G = (y,4); H = (0,x); I = F + z * dir(225); J = G + z * dir(225); draw(A--B--C--D--A); draw(H--E); draw(J--G^^F--I); draw(rightanglemark(G, J, I), linewidth(.5)); draw(rightanglemark(F, I, E), linewidth(.5)); dot("$A$", A, S); dot("$B$", B, S); dot("$C$", C, dir(90)); dot("$D$", D, dir(90)); dot("$E$", E, S); dot("$F$", F, dir(0)); dot("$G$", G, N); dot("$H$", H, W); dot("$I$", I, SW); dot("$J$", J, SW); [/asy]](/static/aops/4892da26cb6334b359f58d4007f89460883779ab.png)
\textbf{(A) } \frac{7}{3} \qquad \textbf{(B) } 8-4\sqrt2 \qquad \textbf{(C) } 1+\sqrt2 \qquad \textbf{(D) } \frac{7}{4}\sqrt2 \qquad \textbf{(E) } 2\sqrt2
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