One of ways is as follows. We start from

Expand[x^2 +  y^2 == (r - 65)^2 && (x - 85)^2 + (y + 185)^2 == (r - 
  55)^2 && (x - 150)^2 + (y + 45)^2 == (r - 35)^2]

x^2 + y^2 == 4225 - 130 r + r^2 && 41450 - 170 x + x^2 + 370 y + y^2 == 3025 - 110 r + r^2 && 24525 - 300 x + x^2 + 90 y + y^2 == 1225 - 70 r + r^2

We see x^2+y^2 in the second and third equations. We simplify those making use of the first equation,

Simplify[ 41450 - 170 x + x^2 + 370 y + y^2 == 3025 - 110 r + r^2 && 
24525 - 300 x + x^2 + 90 y + y^2 == 1225 - 70 r + r^2, 
Assumptions -> x^2 + y^2 == 4225 - 130  r + r^2]

2 r + 17 x == 4265 + 37 y && 4 (r + 5 x) == 1835 + 6 y

Now we can express x and y through r by solving a system of linear equations

Solve[2  r + 17 x == 4265 + 37 y && 4 (r + 5 x) == 1835 + 6  y, {x,  y}]

{x -> 1/638 (42305 - 136 r), y -> 1/638 (-54105 - 28 r)}}

At last, we substitute thes expressions in the first equation and solve it in r.

x^2 + y^2 == (r - 65)^2 /. {x -> 1/638  (42305 - 136  r),  y -> 1/638  (-54105 - 28  r)}

(42305 - 136 r)^2/407044 + (-54105 - 28 r)^2/407044 == (-65 + r)^2

Solve[(42305 - 136 r)^2/407044 + (-54105 - 28 r)^2/407044 == (-65 + r)^2, {r}, Reals]

{{r -> (11109660 - 1595 Sqrt[162728790])/193882}, {r -> ( 11109660 + 1595 Sqrt[162728790])/193882}}

The rest is left on your own.