{"status": "success", "data": {"description_md": "Circles $A, B$ and $C$ are externally tangent to each other, and internally tangent to circle $D$. Circles $B$ and $C$ are congruent. Circle $A$ has radius $1$ and passes through the center of $D$. What is the radius of circle $B$?\n\n<center><img class=\"problem-image\" alt='[asy] unitsize(15mm); pair A=(-1,0),B=(2/3,8/9),C=(2/3,-8/9),D=(0,0);  draw(Circle(D,2)); draw(Circle(A,1)); draw(Circle(B,8/9)); draw(Circle(C,8/9));  label(\"A\", A); label(\"B\", B); label(\"C\", C); label(\"D\", (-1.2,1.8)); [/asy]' class=\"latexcenter\" height=\"285\" src=\"https://latex.artofproblemsolving.com/6/9/6/69677da9f9025d93eca8bc2a4d63d4a3b5b11c9c.png\" width=\"285\"/></center>\n<br>\n\n$\\text{(A) } \\frac23 \\qquad \\text{(B) } \\frac {\\sqrt3}{2} \\qquad \\text{(C) } \\frac78 \\qquad \\text{(D) } \\frac89 \\qquad \\text{(E) } \\frac {1 + \\sqrt3}{3}$\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": "<p>Circles <span class=\"katex--inline\">A, B</span> and <span class=\"katex--inline\">C</span> are externally tangent to each other, and internally tangent to circle <span class=\"katex--inline\">D</span>. Circles <span class=\"katex--inline\">B</span> and <span class=\"katex--inline\">C</span> are congruent. Circle <span class=\"katex--inline\">A</span> has radius <span class=\"katex--inline\">1</span> and passes through the center of <span class=\"katex--inline\">D</span>. What is the radius of circle <span class=\"katex--inline\">B</span>?</p>&#10;<center><img class=\"latexcenter\" alt=\"[asy] unitsize(15mm); pair A=(-1,0),B=(2/3,8/9),C=(2/3,-8/9),D=(0,0);  draw(Circle(D,2)); draw(Circle(A,1)); draw(Circle(B,8/9)); draw(Circle(C,8/9));  label(&#34;A&#34;, A); label(&#34;B&#34;, B); label(&#34;C&#34;, C); label(&#34;D&#34;, (-1.2,1.8)); [/asy]\" height=\"285\" src=\"https://latex.artofproblemsolving.com/6/9/6/69677da9f9025d93eca8bc2a4d63d4a3b5b11c9c.png\" width=\"285\"/></center>&#10;<br/>&#10;<p><span class=\"katex--inline\">\\text{(A) } \\frac23 \\qquad \\text{(B) } \\frac {\\sqrt3}{2} \\qquad \\text{(C) } \\frac78 \\qquad \\text{(D) } \\frac89 \\qquad \\text{(E) } \\frac {1 + \\sqrt3}{3}</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": "2004 AMC 12A Problem 19", "can_next": true, "can_prev": true, "nxt": "/problem/04_amc12A_p20", "prev": "/problem/04_amc12A_p18"}}