{"status": "success", "data": {"description_md": "In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. Distinct points $D$, $E$, and $F$ lie on segments $\\overline{BC}$, $\\overline{CA}$, and $\\overline{DE}$, respectively, such that $\\overline{AD}\\perp\\overline{BC}$, $\\overline{DE}\\perp\\overline{AC}$, and $\\overline{AF}\\perp\\overline{BF}$. The length of segment $\\overline{DF}$ can be written as $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?\n\n$\\textbf{(A)}\\ 18\\qquad\\textbf{(B)}\\ 21\\qquad\\textbf{(C)}\\ 24\\qquad\\textbf{(D)}\\ 27\\qquad\\textbf{(E)}\\ 30$\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 triangle  <span class=\"katex--inline\">ABC</span> ,  <span class=\"katex--inline\">AB=13</span> ,  <span class=\"katex--inline\">BC=14</span> , and  <span class=\"katex--inline\">CA=15</span> . Distinct points  <span class=\"katex--inline\">D</span> ,  <span class=\"katex--inline\">E</span> , and  <span class=\"katex--inline\">F</span>  lie on segments  <span class=\"katex--inline\">\\overline{BC}</span> ,  <span class=\"katex--inline\">\\overline{CA}</span> , and  <span class=\"katex--inline\">\\overline{DE}</span> , respectively, such that  <span class=\"katex--inline\">\\overline{AD}\\perp\\overline{BC}</span> ,  <span class=\"katex--inline\">\\overline{DE}\\perp\\overline{AC}</span> , and  <span class=\"katex--inline\">\\overline{AF}\\perp\\overline{BF}</span> . The length of segment  <span class=\"katex--inline\">\\overline{DF}</span>  can be written as  <span class=\"katex--inline\">\\frac{m}{n}</span> , where  <span class=\"katex--inline\">m</span>  and  <span class=\"katex--inline\">n</span>  are relatively prime positive integers. What is  <span class=\"katex--inline\">m+n</span> ?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A)}\\ 18\\qquad\\textbf{(B)}\\ 21\\qquad\\textbf{(C)}\\ 24\\qquad\\textbf{(D)}\\ 27\\qquad\\textbf{(E)}\\ 30</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": "2013 AMC 12B Problem 19", "can_next": true, "can_prev": true, "nxt": "/problem/13_amc12B_p20", "prev": "/problem/13_amc12B_p18"}}