{"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\"image-2024-07-29-211344557\"\n\n$\\mathrm{(A)}\\ 16\\qquad\\mathrm{(B)}\\ \\frac {50}3\\qquad\\mathrm{(C)}\\ 10\\sqrt{3}\\qquad\\mathrm{(D)}\\ 8\\sqrt{5}\\qquad\\mathrm{(E)}\\ 18$\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": "

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?
\"image-2024-07-29-211344557\"

\\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

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": 5, "problem_name": "2004 AMC 12B Problem 24", "can_next": true, "can_prev": true, "nxt": "/problem/04_amc12B_p25", "prev": "/problem/04_amc12B_p23"}}