{"status": "success", "data": {"description_md": "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 $FI^2$?<br><center><img class=\"problem-image\" alt='[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]' class=\"latexcenter\" height=\"245\" src=\"https://latex.artofproblemsolving.com/4/8/9/4892da26cb6334b359f58d4007f89460883779ab.png\" width=\"252\"/></center>\n\n$\\textbf{(A) } \\frac{7}{3} \\qquad \\textbf{(B) } 8-4\\sqrt2 \\qquad \\textbf{(C) } 1+\\sqrt2 \\qquad \\textbf{(D) } \\frac{7}{4}\\sqrt2 \\qquad \\textbf{(E) } 2\\sqrt2$\n___\nFull credit goes to [MAA](https://maa.org/) for authoring these problems. These problems were taken on the [AOPS](https://artofproblemsolving.com/) website.", "description_html": "<p>In square  <span class=\"katex--inline\">ABCD</span> , points  <span class=\"katex--inline\">E</span>  and  <span class=\"katex--inline\">H</span>  lie on  <span class=\"katex--inline\">\\overline{AB}</span>  and  <span class=\"katex--inline\">\\overline{DA}</span> , respectively, so that  <span class=\"katex--inline\">AE=AH.</span>  Points  <span class=\"katex--inline\">F</span>  and  <span class=\"katex--inline\">G</span>  lie on  <span class=\"katex--inline\">\\overline{BC}</span>  and  <span class=\"katex--inline\">\\overline{CD}</span> , respectively, and points  <span class=\"katex--inline\">I</span>  and  <span class=\"katex--inline\">J</span>  lie on  <span class=\"katex--inline\">\\overline{EH}</span>  so that  <span class=\"katex--inline\">\\overline{FI} \\perp \\overline{EH}</span>  and  <span class=\"katex--inline\">\\overline{GJ} \\perp \\overline{EH}</span> . See the figure below. Triangle  <span class=\"katex--inline\">AEH</span> , quadrilateral  <span class=\"katex--inline\">BFIE</span> , quadrilateral  <span class=\"katex--inline\">DHJG</span> , and pentagon  <span class=\"katex--inline\">FCGJI</span>  each has area  <span class=\"katex--inline\">1.</span>  What is  <span class=\"katex--inline\">FI^2</span> ?<br/><center><img class=\"latexcenter\" alt=\"[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(&#34;$A$&#34;, A, S); dot(&#34;$B$&#34;, B, S); dot(&#34;$C$&#34;, C, dir(90)); dot(&#34;$D$&#34;, D, dir(90)); dot(&#34;$E$&#34;, E, S); dot(&#34;$F$&#34;, F, dir(0)); dot(&#34;$G$&#34;, G, N); dot(&#34;$H$&#34;, H, W); dot(&#34;$I$&#34;, I, SW); dot(&#34;$J$&#34;, J, SW);  [/asy]\" height=\"245\" src=\"https://latex.artofproblemsolving.com/4/8/9/4892da26cb6334b359f58d4007f89460883779ab.png\" width=\"252\"/></center></p>&#10;<p> <span class=\"katex--inline\">\\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</span> </p>&#10;<hr><p>Full credit goes to <a href=\"https://maa.org/\">MAA</a> for authoring these problems. These problems were taken on the <a href=\"https://artofproblemsolving.com/\">AOPS</a> website.</p>", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 3, "problem_name": "2020 AMC 12B Problem 18", "can_next": true, "can_prev": true, "nxt": "/problem/20_amc12B_p19", "prev": "/problem/20_amc12B_p17"}}