Math Problem Statement
In triangle $XYZ,$ circles are drawn centered at $X$, $Y$, and $Z$, so that all pairs of circles are externally tangent. If $XY = 8,$ $XZ = 10,$ and $YZ = 12$, then find the sum of the areas of all three circles.
Asymptote code below [asy] unitsize(1 cm);
pair A, B, C; real[] r;
A = (1,2); B = (0,0); C = (3,0); r[1] = (abs(A - B) + abs(A - C) - abs(B - C))/2; r[2] = (abs(A - B) + abs(B - C) - abs(A - C))/2; r[3] = (abs(A - C) + abs(B - C) - abs(A - B))/2;
draw(A--B--C--cycle); draw(Circle(A,r[1])); draw(Circle(B,r[2])); draw(Circle(C,r[3]));
label("$X$", A, N); label("$Y$", B, SW); label("$Z$", C, SE); [/asy]
Solution
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Math Problem Analysis
Mathematical Concepts
Geometry
System of Equations
Circle Properties
Triangles
Formulas
Sum of circle radii: r_X + r_Y = XY, r_X + r_Z = XZ, r_Y + r_Z = YZ
Area of a circle: A = πr^2
Theorems
External Tangency Condition for Circles
Basic Triangle Geometry
Suitable Grade Level
Grades 9-12
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