{"status": "success", "data": {"description_md": "In $\\triangle ABC$, $AB = BC$, and $\\overline{BD}$ is an [[altitude]]. Point $E$ is on the extension of $\\overline{AC}$ such that $BE = 10$. The values of $\\tan \\angle CBE$, $\\tan \\angle DBE$, and $\\tan \\angle ABE$ form a [[geometric progression]], and the values of $\\cot \\angle DBE,$ $\\cot \\angle CBE,$ $\\cot \\angle DBC$ form an [[arithmetic progression]]. What is the area of $\\triangle ABC$?\n\n$$<center><img class=\"problem-image\" alt='[asy] size(120); defaultpen(0.7); pair A = (0,0), D = (5*2^.5/3,0), C = (10*2^.5/3,0), B = (5*2^.5/3,5*2^.5), E = (13*2^.5/3,0); draw(A--D--C--E--B--C--D--B--cycle); label(\"\\(A\\)\",A,S); label(\"\\(B\\)\",B,N); label(\"\\(C\\)\",C,S); label(\"\\(D\\)\",D,S); label(\"\\(E\\)\",E,S); [/asy]' class=\"latexcenter\" height=\"202\" src=\"https://latex.artofproblemsolving.com/c/5/a/c5ae6e04f0710cb08f0e10ffc47a0f257844910e.png\" width=\"158\"/></center>$$\n\n$\\mathrm{(A)}\\ 16<br>\\qquad\\mathrm{(B)}\\ \\frac {50}3<br>\\qquad\\mathrm{(C)}\\ 10\\sqrt{3}<br>\\qquad\\mathrm{(D)}\\ 8\\sqrt{5}<br>\\qquad\\mathrm{(E)}\\ 18$\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>In  <span class=\"katex--inline\">\\triangle ABC</span> ,  <span class=\"katex--inline\">AB = BC</span> , and  <span class=\"katex--inline\">\\overline{BD}</span>  is an [[altitude]]. Point  <span class=\"katex--inline\">E</span>  is on the extension of  <span class=\"katex--inline\">\\overline{AC}</span>  such that  <span class=\"katex--inline\">BE = 10</span> . The values of  <span class=\"katex--inline\">\\tan \\angle CBE</span> ,  <span class=\"katex--inline\">\\tan \\angle DBE</span> , and  <span class=\"katex--inline\">\\tan \\angle ABE</span>  form a [[geometric progression]], and the values of  <span class=\"katex--inline\">\\cot \\angle DBE,</span>   <span class=\"katex--inline\">\\cot \\angle CBE,</span>   <span class=\"katex--inline\">\\cot \\angle DBC</span>  form an [[arithmetic progression]]. What is the area of  <span class=\"katex--inline\">\\triangle ABC</span> ?</p>&#10;<p> <span class=\"katex--display\">&lt;center&gt;&lt;img class=&#34;problem-image&#34; alt='[asy] size(120); defaultpen(0.7); pair A = (0,0), D = (5*2^.5/3,0), C = (10*2^.5/3,0), B = (5*2^.5/3,5*2^.5), E = (13*2^.5/3,0); draw(A--D--C--E--B--C--D--B--cycle); label(&#34;\\(A\\)&#34;,A,S); label(&#34;\\(B\\)&#34;,B,N); label(&#34;\\(C\\)&#34;,C,S); label(&#34;\\(D\\)&#34;,D,S); label(&#34;\\(E\\)&#34;,E,S); [/asy]' class=&#34;latexcenter&#34; height=&#34;202&#34; src=&#34;https://latex.artofproblemsolving.com/c/5/a/c5ae6e04f0710cb08f0e10ffc47a0f257844910e.png&#34; width=&#34;158&#34;/&gt;&lt;/center&gt;</span> </p>&#10;<p> <span class=\"katex--inline\">\\mathrm{(A)}\\ 16\\qquad\\mathrm{(B)}\\ \\frac {50}3\\qquad\\mathrm{(C)}\\ 10\\sqrt{3}\\qquad\\mathrm{(D)}\\ 8\\sqrt{5}\\qquad\\mathrm{(E)}\\ 18</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": 5, "problem_name": "2004 AMC 12B Problem 24", "can_next": true, "can_prev": true, "nxt": "/problem/04_amc12B_p25", "prev": "/problem/04_amc12B_p23"}}