{"status": "success", "data": {"description_md": "Circles with centers $O$ and $P$ have radii 2 and 4, respectively, and are externally tangent.  Points $A$ and $B$ are on the circle centered at $O$, and points $C$ and $D$ are on the circle centered at $P$, such that $\\overline{AD}$ and $\\overline{BC}$ are common external tangents to the circles.  What is the area of hexagon $AOBCPD$?<br><center><img class=\"problem-image\" alt='[asy] // from amc10 problem series unitsize(0.4 cm); defaultpen(linewidth(0.7) + fontsize(11)); pair A, B, C, D; pair[] O; O[1] = (6,0); O[2] = (12,0); A = (32/6,8*sqrt(2)/6); B = (32/6,-8*sqrt(2)/6); C = 2*B; D = 2*A; draw(Circle(O[1],2)); draw(Circle(O[2],4)); draw((0.7*A)--(1.2*D)); draw((0.7*B)--(1.2*C)); draw(O[1]--O[2]); draw(A--O[1]); draw(B--O[1]); draw(C--O[2]); draw(D--O[2]); label(\"$A$\", A, NW); label(\"$B$\", B, SW); label(\"$C$\", C, SW); label(\"$D$\", D, NW); dot(\"$O$\", O[1], SE); dot(\"$P$\", O[2], SE); label(\"$2$\", (A + O[1])/2, E); label(\"$4$\", (D + O[2])/2, E);[/asy]' class=\"latexcenter\" height=\"182\" src=\"https://latex.artofproblemsolving.com/9/8/5/985d18b9bd1bc04f16562ee9a3eabc5afe6efcbd.png\" width=\"235\"/></center>\n\n$\\textbf{(A) } 18\\sqrt {3} \\qquad \\textbf{(B) } 24\\sqrt {2} \\qquad \\textbf{(C) } 36 \\qquad \\textbf{(D) } 24\\sqrt {3} \\qquad \\textbf{(E) } 32\\sqrt {2}$\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 with centers  <span class=\"katex--inline\">O</span>  and  <span class=\"katex--inline\">P</span>  have radii 2 and 4, respectively, and are externally tangent.  Points  <span class=\"katex--inline\">A</span>  and  <span class=\"katex--inline\">B</span>  are on the circle centered at  <span class=\"katex--inline\">O</span> , and points  <span class=\"katex--inline\">C</span>  and  <span class=\"katex--inline\">D</span>  are on the circle centered at  <span class=\"katex--inline\">P</span> , such that  <span class=\"katex--inline\">\\overline{AD}</span>  and  <span class=\"katex--inline\">\\overline{BC}</span>  are common external tangents to the circles.  What is the area of hexagon  <span class=\"katex--inline\">AOBCPD</span> ?<br/><center><img class=\"latexcenter\" alt=\"[asy] // from amc10 problem series unitsize(0.4 cm); defaultpen(linewidth(0.7) + fontsize(11)); pair A, B, C, D; pair[] O; O[1] = (6,0); O[2] = (12,0); A = (32/6,8*sqrt(2)/6); B = (32/6,-8*sqrt(2)/6); C = 2*B; D = 2*A; draw(Circle(O[1],2)); draw(Circle(O[2],4)); draw((0.7*A)--(1.2*D)); draw((0.7*B)--(1.2*C)); draw(O[1]--O[2]); draw(A--O[1]); draw(B--O[1]); draw(C--O[2]); draw(D--O[2]); label(&#34;$A$&#34;, A, NW); label(&#34;$B$&#34;, B, SW); label(&#34;$C$&#34;, C, SW); label(&#34;$D$&#34;, D, NW); dot(&#34;$O$&#34;, O[1], SE); dot(&#34;$P$&#34;, O[2], SE); label(&#34;$2$&#34;, (A + O[1])/2, E); label(&#34;$4$&#34;, (D + O[2])/2, E);[/asy]\" height=\"182\" src=\"https://latex.artofproblemsolving.com/9/8/5/985d18b9bd1bc04f16562ee9a3eabc5afe6efcbd.png\" width=\"235\"/></center></p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A) } 18\\sqrt {3} \\qquad \\textbf{(B) } 24\\sqrt {2} \\qquad \\textbf{(C) } 36 \\qquad \\textbf{(D) } 24\\sqrt {3} \\qquad \\textbf{(E) } 32\\sqrt {2}</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 12B Problem 15", "can_next": true, "can_prev": true, "nxt": "/problem/06_amc12B_p16", "prev": "/problem/06_amc12B_p14"}}