Right triangle with vertices as centers of circles
Source: AMC 12 2006A, Problem 13
February 5, 2006
geometryalgebrasystem of equationsAMC
Problem Statement
The vertices of a 3 \minus{} 4 \minus{} 5 right triangle are the centers of three mutually externally tangent circles, as shown. What is the sum of the areas of the three circles?
[asy]unitsize(5mm);
defaultpen(fontsize(10pt)+linewidth(.8pt));pair B=(0,0), C=(5,0);
pair A=intersectionpoints(Circle(B,3),Circle(C,4))[0];draw(A--B--C--cycle);
draw(Circle(C,3));
draw(Circle(A,1));
draw(Circle(B,2));label("",A,N);
label("",B,W);
label("",C,E);
label("3",midpoint(B--A),NW);
label("4",midpoint(A--C),NE);
label("5",midpoint(B--C),S);[/asy]