{"status": "success", "data": {"description_md": "On square $ABCD,$ points $E,F,G,$ and $H$ lie on sides $\\overline{AB},\\overline{BC},\\overline{CD},$ and $\\overline{DA},$ respectively, so that $\\overline{EG} \\perp \\overline{FH}$ and $EG=FH = 34.$ Segments $\\overline{EG}$ and $\\overline{FH}$ intersect at a point $P,$ and the areas of the quadrilaterals $AEPH, BFPE, CGPF,$ and $DHPG$ are in the ratio $269:275:405:411.$ Find the area of square $ABCD$.<br><br><br>$\\includegraphics[width=154, height=167, totalheight=167]{https://latex.artofproblemsolving.com/a/0/a/a0abaa3752c254ff6e849c7c2fd86091c343b21a.png}$\n___\nLeading zeroes must be inputted, so if your answer is `34`, then input `034`. Full 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>On square <span class=\"katex--inline\">ABCD,</span> points <span class=\"katex--inline\">E,F,G,</span> and <span class=\"katex--inline\">H</span> lie on sides <span class=\"katex--inline\">\\overline{AB},\\overline{BC},\\overline{CD},</span> and <span class=\"katex--inline\">\\overline{DA},</span> respectively, so that <span class=\"katex--inline\">\\overline{EG} \\perp \\overline{FH}</span> and <span class=\"katex--inline\">EG=FH = 34.</span> Segments <span class=\"katex--inline\">\\overline{EG}</span> and <span class=\"katex--inline\">\\overline{FH}</span> intersect at a point <span class=\"katex--inline\">P,</span> and the areas of the quadrilaterals <span class=\"katex--inline\">AEPH, BFPE, CGPF,</span> and <span class=\"katex--inline\">DHPG</span> are in the ratio <span class=\"katex--inline\">269:275:405:411.</span> Find the area of square <span class=\"katex--inline\">ABCD</span>.<br/><br/><br/><img src=\"https://latex.artofproblemsolving.com/a/0/a/a0abaa3752c254ff6e849c7c2fd86091c343b21a.png\" width=\"154\" height=\"167\" class=\"problem-image\"/></p>&#10;<hr><p>Leading zeroes must be inputted, so if your answer is <code>34</code>, then input <code>034</code>. 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": 6, "problem_name": "2014 AIME I Problem 13", "can_next": true, "can_prev": true, "nxt": "/problem/14_aime_I_p14", "prev": "/problem/14_aime_I_p12"}}