{"status": "success", "data": {"description_md": "Quadrilateral $ABCD$ is inscribed in circle $O$ and has side lengths $AB=3, BC=2, CD=6$, and $DA=8$. Let $X$ and $Y$ be points on $\\overline{BD}$ such that $\\frac{DX}{BD} = \\frac{1}{4}$ and $\\frac{BY}{BD} = \\frac{11}{36}$. Let $E$ be the intersection of line $AX$ and the line through $Y$ parallel to $\\overline{AD}$. Let $F$ be the intersection of line $CX$ and the line through $E$ parallel to $\\overline{AC}$. Let $G$ be the point on circle $O$ other than $C$ that lies on line $CX$. What is $XF\\cdot XG$?\n\n$\\textbf{(A) }17\\qquad\\textbf{(B) }\\frac{59 - 5\\sqrt{2}}{3}\\qquad\\textbf{(C) }\\frac{91 - 12\\sqrt{3}}{4}\\qquad\\textbf{(D) }\\frac{67 - 10\\sqrt{2}}{3}\\qquad\\textbf{(E) }18$\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>Quadrilateral  <span class=\"katex--inline\">ABCD</span>  is inscribed in circle  <span class=\"katex--inline\">O</span>  and has side lengths  <span class=\"katex--inline\">AB=3, BC=2, CD=6</span> , and  <span class=\"katex--inline\">DA=8</span> . Let  <span class=\"katex--inline\">X</span>  and  <span class=\"katex--inline\">Y</span>  be points on  <span class=\"katex--inline\">\\overline{BD}</span>  such that  <span class=\"katex--inline\">\\frac{DX}{BD} = \\frac{1}{4}</span>  and  <span class=\"katex--inline\">\\frac{BY}{BD} = \\frac{11}{36}</span> . Let  <span class=\"katex--inline\">E</span>  be the intersection of line  <span class=\"katex--inline\">AX</span>  and the line through  <span class=\"katex--inline\">Y</span>  parallel to  <span class=\"katex--inline\">\\overline{AD}</span> . Let  <span class=\"katex--inline\">F</span>  be the intersection of line  <span class=\"katex--inline\">CX</span>  and the line through  <span class=\"katex--inline\">E</span>  parallel to  <span class=\"katex--inline\">\\overline{AC}</span> . Let  <span class=\"katex--inline\">G</span>  be the point on circle  <span class=\"katex--inline\">O</span>  other than  <span class=\"katex--inline\">C</span>  that lies on line  <span class=\"katex--inline\">CX</span> . What is  <span class=\"katex--inline\">XF\\cdot XG</span> ?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A) }17\\qquad\\textbf{(B) }\\frac{59 - 5\\sqrt{2}}{3}\\qquad\\textbf{(C) }\\frac{91 - 12\\sqrt{3}}{4}\\qquad\\textbf{(D) }\\frac{67 - 10\\sqrt{2}}{3}\\qquad\\textbf{(E) }18</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": 5, "problem_name": "2017 AMC 12A Problem 24", "can_next": true, "can_prev": true, "nxt": "/problem/17_amc12A_p25", "prev": "/problem/17_amc12A_p23"}}