{"status": "success", "data": {"description_md": "Let $AB$ be a diameter of a circle and let $C$ be a point on $AB$ with $2\\cdot AC=BC$. Let $D$ and $E$ be points on the circle such that $DC \\perp AB$ and $DE$ is a second diameter. What is the ratio of the area of $\\triangle DCE$ to the area of $\\triangle ABD$?\n\n<center>\n<img class=\"problem-image\" height=\"262\" src=\"https://latex.artofproblemsolving.com/a/8/0/a805b2766042251797b5817c889a0fb79a637874.png\" width=\"272\"/>\n</center><br>\n\n$\\mathrm{(A) \\ } \\frac{1}{6} \\qquad \\mathrm{(B) \\ } \\frac{1}{4} \\qquad \\mathrm{(C) \\ } \\frac{1}{3} \\qquad \\mathrm{(D) \\ } \\frac{1}{2} \\qquad \\mathrm{(E) \\ } \\frac{2}{3}$", "description_html": "<p>Let <span class=\"katex--inline\">AB</span> be a diameter of a circle and let <span class=\"katex--inline\">C</span> be a point on <span class=\"katex--inline\">AB</span> with <span class=\"katex--inline\">2\\cdot AC=BC</span>. Let <span class=\"katex--inline\">D</span> and <span class=\"katex--inline\">E</span> be points on the circle such that <span class=\"katex--inline\">DC \\perp AB</span> and <span class=\"katex--inline\">DE</span> is a second diameter. What is the ratio of the area of <span class=\"katex--inline\">\\triangle DCE</span> to the area of <span class=\"katex--inline\">\\triangle ABD</span>?</p>&#10;<center>&#10;<img class=\"problem-image\" height=\"262\" src=\"https://latex.artofproblemsolving.com/a/8/0/a805b2766042251797b5817c889a0fb79a637874.png\" width=\"272\"/>&#10;</center><br/>&#10;<p><span class=\"katex--inline\">\\mathrm{(A) \\ } \\frac{1}{6} \\qquad \\mathrm{(B) \\ } \\frac{1}{4} \\qquad \\mathrm{(C) \\ } \\frac{1}{3} \\qquad \\mathrm{(D) \\ } \\frac{1}{2} \\qquad \\mathrm{(E) \\ } \\frac{2}{3}</span></p>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 4, "problem_name": "2005 AMC 10A Problem 23", "can_next": true, "can_prev": true, "nxt": "/problem/05_amc10A_p24", "prev": "/problem/05_amc10A_p22"}}