Internal tangent of circles
Source: AMC 12 2006A, Problem 16
February 5, 2006
ratioAMCAMC 10AMC 12Pythagorean Theoremgeometry
Problem Statement
Circles with centers and have radii 3 and 8, respectively. A common internal tangent intersects the circles at and , respectively. Lines and intersect at , and AE \equal{} 5. What is ?
[asy]unitsize(2.5mm);
defaultpen(fontsize(10pt)+linewidth(.8pt));
dotfactor=3;pair A=(0,0), Ep=(5,0), B=(5+40/3,0);
pair M=midpoint(A--Ep);
pair C=intersectionpoints(Circle(M,2.5),Circle(A,3))[1];
pair D=B+8*dir(180+degrees(C));dot(A);
dot(C);
dot(B);
dot(D);
draw(C--D);
draw(A--B);
draw(Circle(A,3));
draw(Circle(B,8));label("",A,W);
label("",B,E);
label("",C,SE);
label("",Ep,SSE);
label("",D,NW);[/asy]