If xy=18, yz=14 and xz=20, find the radius of each circle.

Let`s assume that points M, N and P are the touching points of those 3 circles:
Then:
Y M + M Z = 14,
Z N + N X = 20
X P + P Y = 18
And also: M Z = ZN, Y M = P Y and N X = X P.
Now we have a system of 3 equations ( Y M, M Z and X P  are the radii of each circle ):
Y M + M Z = 14
M Z + X P = 20
X P + Y M = 18

- Y M - M Z = - 14
+
X P  + Y M = 18

X P - M Z  = 4

- Y M - M Z = - 14
+
M Z + X P = 20

X P - Y M = 6  /* ( - 1 )
X P  - M Z = 4  

- X P + Y M = - 6
  X P - M Z = 4

           Y M - M Z = - 2
           Y M + M Z = 14
         
           2 Y M = 12  =>  Y M = 6
M Z - 6 = 2  =>   M Z = 8
X P + 6 = 18  => X P = 12
Radii of the circles are: 12, 8 and 6.  
 

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