{"status": "success", "data": {"description_md": "Triangle $ABC$ has $AB = 13, BC = 14$, and $AC = 15$. The points $D, E$, and $F$ are the midpoints of $\\overline{AB}, \\overline{BC}$, and $\\overline{AC}$ respectively. Let $X \\neq E$ be the intersection of the circumcircles of $\\Delta BDE$ and $\\Delta CEF$. What is $XA + XB + XC$?\n\n$\\textbf{(A)}\\ 24 \\qquad \\textbf{(B)}\\ 14\\sqrt{3} \\qquad \\textbf{(C)}\\ \\frac{195}{8} \\qquad \\textbf{(D)}\\ \\frac{129\\sqrt{7}}{14} \\qquad \\textbf{(E)}\\ \\frac{69\\sqrt{2}}{4}$\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>Triangle <span class=\"katex--inline\">ABC</span> has <span class=\"katex--inline\">AB = 13, BC = 14</span>, and <span class=\"katex--inline\">AC = 15</span>. The points <span class=\"katex--inline\">D, E</span>, and <span class=\"katex--inline\">F</span> are the midpoints of <span class=\"katex--inline\">\\overline{AB}, \\overline{BC}</span>, and <span class=\"katex--inline\">\\overline{AC}</span> respectively. Let <span class=\"katex--inline\">X \\neq E</span> be the intersection of the circumcircles of <span class=\"katex--inline\">\\Delta BDE</span> and <span class=\"katex--inline\">\\Delta CEF</span>. What is <span class=\"katex--inline\">XA + XB + XC</span>?</p>&#10;<p><span class=\"katex--inline\">\\textbf{(A)}\\ 24 \\qquad \\textbf{(B)}\\ 14\\sqrt{3} \\qquad \\textbf{(C)}\\ \\frac{195}{8} \\qquad \\textbf{(D)}\\ \\frac{129\\sqrt{7}}{14} \\qquad \\textbf{(E)}\\ \\frac{69\\sqrt{2}}{4}</span></p>&#10;<hr/>&#10;<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>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 3, "problem_name": "2011 AMC 12B Problem 20", "can_next": true, "can_prev": true, "nxt": "/problem/11_amc12B_p21", "prev": "/problem/11_amc12B_p19"}}