{"status": "success", "data": {"description_md": "Let $\\overline{AB}$ be a diameter of a circle and $C$ be a point on $\\overline{AB}$ with $2 \\cdot AC = BC$. Let $D$ and $E$ be points on the circle such that $\\overline{DC} \\perp \\overline{AB}$ and $\\overline{DE}$ is a second diameter. What is the ratio of the area of $\\triangle DCE$ to the area of $\\triangle ABD$?
[asy] unitsize(2.5cm); defaultpen(fontsize(10pt)+linewidth(.8pt)); dotfactor=3; pair O=(0,0), C=(-1/3.0), B=(1,0), A=(-1,0); pair D=dir(aCos(C.x)), E=(-D.x,-D.y); draw(A--B--D--cycle); draw(D--E--C); draw(unitcircle,white); drawline(D,C); dot(O); clip(unitcircle); draw(unitcircle); label(\"$E$\",E,SSE); label(\"$B$\",B,E); label(\"$A$\",A,W); label(\"$D$\",D,NNW); label(\"$C$\",C,SW); draw(rightanglemark(D,C,B,2));[/asy]
\n\n$(\\text {A}) \\ \\frac {1}{6} \\qquad (\\text {B}) \\ \\frac {1}{4} \\qquad (\\text {C})\\ \\frac {1}{3} \\qquad (\\text {D}) \\ \\frac {1}{2} \\qquad (\\text {E})\\ \\frac {2}{3}$\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": "

Let \\overline{AB} be a diameter of a circle and C be a point on \\overline{AB} with 2 \\cdot AC = BC . Let D and E be points on the circle such that \\overline{DC} \\perp \\overline{AB} and \\overline{DE} is a second diameter. What is the ratio of the area of \\triangle DCE to the area of \\triangle ABD ?

\"[asy]

(\\text {A}) \\ \\frac {1}{6} \\qquad (\\text {B}) \\ \\frac {1}{4} \\qquad (\\text {C})\\ \\frac {1}{3} \\qquad (\\text {D}) \\ \\frac {1}{2} \\qquad (\\text {E})\\ \\frac {2}{3}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.

", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 3, "problem_name": "2005 AMC 12A Problem 15", "can_next": true, "can_prev": true, "nxt": "/problem/05_amc12A_p16", "prev": "/problem/05_amc12A_p14"}}