Newbie and systems of quadratic equations

There are "better" ways to solve my problem.
BUT, part of my goal is to refresh some math I haven't seen for almost 
half a century.

I have a system of equations
(x-a1)^2 +(y-b1)^2 +(z-c1)^2 = r1^2
(x-a2)^2 +(y-b2)^2 +(z-c2)^2 = r2^2
(x-a3)^2 +(y-b3)^2 +(z-c3)^2 = r3^2

which expand to
x^2-2*a1*x+a1^2 +y^2-2*b1*y+b1^2 +z^2-2*c1*z+c1^2 = r1^2
x^2-2*a2*x+a2^2 +y^2-2*b2*y+b2^2 +z^2-2*c2*z+c2^2 = r2^2
x^2-2*a3*x+a3^2 +y^2-2*b3*y+b3^2 +z^2-2*c3*z+c3^2 = r3^2

[The a's, b's, c's, and r's for each problem are constant but will be 
different from problem to problem, so I wish a symbolic solution.]

These can be rewritten as
f1(x,y,z)=r1^2 -r2^2
f2(x,y,z)=r2^2 -r3^2
f3(x,y,z)=r3^2 -r1^2

which are convienently linear in x, y, and z.

Can Maxima do that subtraction for me?

I can do it with paper and pencil but that is prone to copy errors and 
legibility. Also eventually I'll have to enter into the computer any way.