{"status": "success", "data": {"description_md": "In $\\triangle ABC$, $AB = 6$, $BC = 7$, and $CA = 8$. Point $D$ lies on $\\overline{BC}$, and $\\overline{AD}$ bisects $\\angle BAC$. Point $E$ lies on $\\overline{AC}$, and $\\overline{BE}$ bisects $\\angle ABC$. The bisectors intersect at $F$. What is the ratio $AF$ : $FD$?
[asy] pair A = (0,0), B=(6,0), C=intersectionpoints(Circle(A,8),Circle(B,7))[0], F=incenter(A,B,C), D=extension(A,F,B,C),E=extension(B,F,A,C); draw(A--B--C--A--D^^B--E); label(\"$A$\",A,SW); label(\"$B$\",B,SE); label(\"$C$\",C,N); label(\"$D$\",D,NE); label(\"$E$\",E,NW); label(\"$F$\",F,1.5*N); [/asy]
\n\n$\\textbf{(A)}\\ 3:2\\qquad\\textbf{(B)}\\ 5:3\\qquad\\textbf{(C)}\\ 2:1\\qquad\\textbf{(D)}\\ 7:3\\qquad\\textbf{(E)}\\ 5:2$\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": "

In \\triangle ABC , AB = 6 , BC = 7 , and CA = 8 . Point D lies on \\overline{BC} , and \\overline{AD} bisects \\angle BAC . Point E lies on \\overline{AC} , and \\overline{BE} bisects \\angle ABC . The bisectors intersect at F . What is the ratio AF : FD ?

\"[asy]

\\textbf{(A)}\\ 3:2\\qquad\\textbf{(B)}\\ 5:3\\qquad\\textbf{(C)}\\ 2:1\\qquad\\textbf{(D)}\\ 7:3\\qquad\\textbf{(E)}\\ 5:2

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": "2016 AMC 12A Problem 12", "can_next": true, "can_prev": true, "nxt": "/problem/16_amc12A_p13", "prev": "/problem/16_amc12A_p11"}}