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2015 Math Prize for Girls Problems
20
20
Part of
2015 Math Prize for Girls Problems
Problems
(1)
Math Prize 2015 Problem 20
Source:
9/23/2015
In the diagram below, the circle with center
A
A
A
is congruent to and tangent to the circle with center
B
B
B
. A third circle is tangent to the circle with center
A
A
A
at point
C
C
C
and passes through point
B
B
B
. Points
C
C
C
,
A
A
A
, and
B
B
B
are collinear. The line segment
C
D
E
F
G
‾
\overline{CDEFG}
C
D
EFG
intersects the circles at the indicated points. Suppose that
D
E
=
6
DE = 6
D
E
=
6
and
F
G
=
9
FG = 9
FG
=
9
. Find
A
G
AG
A
G
. [asy] unitsize(5); pair A = (-9 sqrt(3), 0); pair B = (9 sqrt(3), 0); pair C = (-18 sqrt(3), 0); pair D = (-4 sqrt(3) / 3, 10 sqrt(6) / 3); pair E = (2 sqrt(3), 4 sqrt(6)); pair F = (7 sqrt(3), 5 sqrt(6)); pair G = (12 sqrt(3), 6 sqrt(6)); real r = 9sqrt(3); draw(circle(A, r)); draw(circle(B, r)); draw(circle((B + C) / 2, 3r / 2)); draw(C -- D); draw("
6
6
6
", E -- D); draw(E -- F); draw("
9
9
9
", F -- G); dot(A); dot(B); label("
A
A
A
", A, plain.E); label("
B
B
B
", B, plain.E); label("
C
C
C
", C, W); label("
D
D
D
", D, dir(160)); label("
E
E
E
", E, S); label("
F
F
F
", F, SSW); label("
G
G
G
", G, N); [/asy]