{"status": "success", "data": {"description_md": "Equilateral triangle $ABC$ has side length $840$. Point $D$ lies on the same side of line $BC$ as $A$ such that $\\overline{BD} \\perp \\overline{BC}$. The line $\\ell$ through $D$ parallel to line $BC$ intersects sides $\\overline{AB}$ and $\\overline{AC}$ at points $E$ and $F$, respectively. Point $G$ lies on $\\ell$ such that $F$ is between $E$ and $G$, $\\triangle AFG$ is isosceles, and the ratio of the area of $\\triangle AFG$ to the area of $\\triangle BED$ is $8:9$. Find $AF$.

$\\includegraphics[width=126, height=101, totalheight=101]{https://latex.artofproblemsolving.com/7/1/5/7154e7a32b3eda0a8ba22787a8b4d10ba8b8dc50.png}$\n___\nLeading zeroes must be inputted, so if your answer is `34`, then input `034`. Full credit goes to [MAA](https://maa.org/) for authoring these problems. These problems were taken on the [AOPS](https://artofproblemsolving.com/) website.", "description_html": "

Equilateral triangle ABC has side length 840. Point D lies on the same side of line BC as A such that \\overline{BD} \\perp \\overline{BC}. The line \\ell through D parallel to line BC intersects sides \\overline{AB} and \\overline{AC} at points E and F, respectively. Point G lies on \\ell such that F is between E and G, \\triangle AFG is isosceles, and the ratio of the area of \\triangle AFG to the area of \\triangle BED is 8:9. Find AF.

Leading zeroes must be inputted, so if your answer is 34, then input 034. 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": 3, "problem_name": "2021 AIME II Problem 2", "can_next": true, "can_prev": true, "nxt": "/problem/21_aime_II_p03", "prev": "/problem/21_aime_II_p01"}}