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MAA AMC
AMC 12/AHSME
1982 AMC 12/AHSME
24
24
Part of
1982 AMC 12/AHSME
Problems
(1)
Circle and Triangle
Source:
1/19/2006
In the adjoining figure, the circle meets the sides of an equilateral triangle at six points. If
A
G
=
2
AG=2
A
G
=
2
,
G
F
=
13
GF=13
GF
=
13
,
F
C
=
1
FC=1
FC
=
1
, and
H
J
=
7
HJ=7
H
J
=
7
, then
D
E
DE
D
E
equals [asy] size(200); defaultpen(fontsize(10)); real r=sqrt(22); pair B=origin, A=16*dir(60), C=(16,0), D=(10-r,0), E=(10+r,0), F=C+1*dir(120), G=C+14*dir(120), H=13*dir(60), J=6*dir(60), O=circumcenter(G,H,J); dot(A^^B^^C^^D^^E^^F^^G^^H^^J); draw(Circle(O, abs(O-D))^^A--B--C--cycle, linewidth(0.7)); label("
A
A
A
", A, N); label("
B
B
B
", B, dir(210)); label("
C
C
C
", C, dir(330)); label("
D
D
D
", D, SW); label("
E
E
E
", E, SE); label("
F
F
F
", F, dir(170)); label("
G
G
G
", G, dir(250)); label("
H
H
H
", H, SE); label("
J
J
J
", J, dir(0)); label("2", A--G, dir(30)); label("13", F--G, dir(180+30)); label("1", F--C, dir(30)); label("7", H--J, dir(-30));[/asy]
(A)
2
22
(B)
7
3
(C)
9
(D)
10
(E)
13
\textbf {(A) } 2\sqrt{22} \qquad \textbf {(B) } 7\sqrt{3} \qquad \textbf {(C) } 9 \qquad \textbf {(D) } 10 \qquad \textbf {(E) } 13
(A)
2
22
(B)
7
3
(C)
9
(D)
10
(E)
13
geometry
power of a point