{"status": "success", "data": {"description_md": "The vertices of a $3-4-5$ right triangle are the centers of three mutually externally tangent circles, as shown. What is the sum of the areas of the three circles? <br><center><img  class=\"problem-image\"  alt=\"[asy] unitsize(5mm); defaultpen(fontsize(10pt)+linewidth(.8pt)); pair B=(0,0), C=(5,0); pair A=intersectionpoints(Circle(B,3),Circle(C,4))[0]; draw(A--B--C--cycle); draw(Circle(C,3)); draw(Circle(A,1)); draw(Circle(B,2)); label(&quot;$A$&quot;,A,N); label(&quot;$B$&quot;,B,W); label(&quot;$C$&quot;,C,E); label(&quot;3&quot;,midpoint(B--A),NW); label(&quot;4&quot;,midpoint(A--C),NE); label(&quot;5&quot;,midpoint(B--C),S);[/asy]\"  class=\"latexcenter\"  height=\"155\"  src=\"https://latex.artofproblemsolving.com/c/d/7/cd70a3d199aac3650e14fb5d9e5786581c4fa2ba.png\"  width=\"238\"/></center>\n\n$\\mathrm{(A) \\ } 12\\pi\\qquad  \\mathrm{(B) \\ } \\frac{25\\pi}{2}\\qquad  \\mathrm{(C) \\ } 13\\pi\\qquad  \\mathrm{(D) \\ } \\frac{27\\pi}{2}\\qquad  \\mathrm{(E) \\ } 14\\pi$\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>The vertices of a  <span class=\"katex--inline\">3-4-5</span>  right triangle are the centers of three mutually externally tangent circles, as shown. What is the sum of the areas of the three circles? <br/><center><img class=\"latexcenter\" alt=\"[asy] unitsize(5mm); defaultpen(fontsize(10pt)+linewidth(.8pt)); pair B=(0,0), C=(5,0); pair A=intersectionpoints(Circle(B,3),Circle(C,4))[0]; draw(A--B--C--cycle); draw(Circle(C,3)); draw(Circle(A,1)); draw(Circle(B,2)); label(&#34;$A$&#34;,A,N); label(&#34;$B$&#34;,B,W); label(&#34;$C$&#34;,C,E); label(&#34;3&#34;,midpoint(B--A),NW); label(&#34;4&#34;,midpoint(A--C),NE); label(&#34;5&#34;,midpoint(B--C),S);[/asy]\" height=\"155\" src=\"https://latex.artofproblemsolving.com/c/d/7/cd70a3d199aac3650e14fb5d9e5786581c4fa2ba.png\" width=\"238\"/></center></p>&#10;<p> <span class=\"katex--inline\">\\mathrm{(A) \\ } 12\\pi\\qquad  \\mathrm{(B) \\ } \\frac{25\\pi}{2}\\qquad  \\mathrm{(C) \\ } 13\\pi\\qquad  \\mathrm{(D) \\ } \\frac{27\\pi}{2}\\qquad  \\mathrm{(E) \\ } 14\\pi</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": 3, "problem_name": "2006 AMC 12A Problem 13", "can_next": true, "can_prev": true, "nxt": "/problem/06_amc12A_p14", "prev": "/problem/06_amc12A_p12"}}