Great job, Tilly! I just got it over here and then checked "Newest Nodes" before posting.
You beat me to the punch!

I'll try to transform my coordinates into yours for this post....

<UPDATE>After all the excitement of thinking I had a different way of acheiving the solution than Tilly, it turns out it's basically the same. I'm leaving the post alive because it may be nice to have the solution to that final quadratic equation, which I include at the end.</UPDATE>

Once getting z as you did, I plugged back into the original equation (x-x')^2+(y-y')^2=(z-z')^2/m^2 using each point for (x',y') and h1..h4 for z'. This gave me four equations (xx means x^2):

xx-2x+1+yy     =aa/mm
xx+2x+1+yy     =cc/mm
xx     +yy-2y+1=bb/mm
xx     +yy+2y+1=dd/mm
[download]

I subtracted the first two to get 4*x*m*m=h3^2-h1^2.
I subtracted the last two to get 4*y*m*m=h4^2-h2^2.
Dividing these two values: 4*y*m*m/(4*x*m*m)=(h4^2-h2^2)/(h3^2-h1^2)
Simplifying: y/x=(h4^2-h2^2)/(h3^2-h1^2)
Then I plugged back into the original equation with just one point (1,0):

Let k=y/x=(h4^2-h2^2)/(h3^2-h1^2) (from above)
Let n=x*m*m=(h3^2-h1^2)/4         (from above)

Original equation:
(x-x')^2+(y-y')^2=(z-z')^2/m^2
(x-1 )^2+(y-0 )^2=(h1  )^2/m^2
(x-1 )^2+(xk-0)^2=(h1  )^2*x/n
(x-1 )^2+(xk-0)^2=(h1  )^2*x/n
multiplying it out:
x^2-2x+1+x^2*k^2 =x*h1^2/n
combining terms:
x^2*(1+k^2)-x(2+h1^2/n)+1=0

let a=1+k^2
let b=2+h1^2/n
x1=(b+sqrt(b^2-4a))/2a
x2=(b-sqrt(b^2-4a))/2a
y1=k*x1
y2=k*x2
[download]

Sadly, only one of (x1,y1) or (x2,y2) is a valid solution.