Tangent Circles
Source: AMC 12 2006B, Problem 15
February 17, 2006
geometryrectangleAMC
Problem Statement
Circles with centers and have radii 2 and 4, respectively, and are externally tangent. Points and are on the circle centered at , and points and are on the circle centered at , such that and are common external tangents to the circles. What is the area of hexagon ?[asy]
unitsize(0.4 cm); defaultpen(linewidth(0.7) + fontsize(11));
pair A, B, C, D;
pair[] O;
O[1] = (6,0);
O[2] = (12,0);
A = (32/6,8*sqrt(2)/6);
B = (32/6,-8*sqrt(2)/6);
C = 2*B;
D = 2*A;
draw(Circle(O[1],2));
draw(Circle(O[2],4));
draw((0.7*A)--(1.2*D));
draw((0.7*B)--(1.2*C));
draw(O[1]--O[2]);
draw(A--O[1]);
draw(B--O[1]);
draw(C--O[2]);
draw(D--O[2]);
label("", A, NW);
label("", B, SW);
label("", C, SW);
label("", D, NW);
dot("", O[1], SE);
dot("", O[2], SE);
label("", (A + O[1])/2, E);
label("", (D + O[2])/2, E);[/asy]