Question: `eliminate` returns nothing?

The goal is to eliminate x, y and z from [a^2=(4*y*z)/((x+y)*(x+z)),b^2=(4*z*x)/((y+z)*(y+x)),c^2=(4*x*y)/((z+x)*(z+y))]. However, eliminate only outputs a null expression (I added a {} to emphasize it): 

restart;
expr := 4*[y*z/((x + y)*(x + z)), z*x/((y + z)*(y + x)), x*y/((z + x)*(z + y))]:
{eliminate}([a, b, c]**~2 =~ expr, [x, y, z]);
 = 
                               {}

Why is the result empty? 
In my view, the result should be (a*b*c)**2 = ((a**2 + b**2 + c**2) - 2**2)**2 (or its equivalent). One may verify this by: 

seq(seq(seq(
    is(eval((a^2 + b^2 + c^2 - 4)^2 = (a*b*c)^2, 
      elementwise([a, b, c] = [k1, k2, k3]*sqrt(expr)))), 
    `in`(k3, [-1, +1])), `in`(k2, [-1, +1])), `in`(k1, [-1, +1]));
 = 
         true, true, true, true, true, true, true, true

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