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Perhaps you have too many parameters. Maybe some more concrete approach can be moe successful:

var('x2 y2')
Z=[1,2]
for x0 in Z:
  for y0 in Z:
    for x1 in Z:
      for y1 in Z:
        for r0 in Z:
          for r1 in Z:
            eq1 = (x2 - x0)^2 + (y2 - y0)^2 == r0^2    
            eq2 = (x2 - x1)^2 + (y2 - y1)^2 == r1^2
            print solve([eq1,eq2], [x2, y2])

Perhaps you have too many parameters. Maybe some more concrete approach can be moe more successful:

var('x2 y2')
Z=[1,2]
for x0 in Z:
  for y0 in Z:
    for x1 in Z:
      for y1 in Z:
        for r0 in Z:
          for r1 in Z:
            eq1 = (x2 - x0)^2 + (y2 - y0)^2 == r0^2    
            eq2 = (x2 - x1)^2 + (y2 - y1)^2 == r1^2
            print solve([eq1,eq2], [x2, y2])