MathDB

In the diagram below, the circle with center

A

is congruent to and tangent to the circle with center

B

. A third circle is tangent to the circle with center

A

at point

C

and passes through point

B

. Points

C

A

, and

B

are collinear. The line segment

\overline{CDEFG}

intersects the circles at the indicated points. Suppose that

DE = 6

and

FG = 9

. Find

AG

. [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

", E -- D); draw(E -- F); draw("

9

", F -- G); dot(A); dot(B); label("

A

", A, plain.E); label("

B

", B, plain.E); label("

C

", C, W); label("

D

", D, dir(160)); label("

E

", E, S); label("

F

", F, SSW); label("

G

", G, N); [/asy]

Problem Statement