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.

[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]