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
Answer: Radii of the circles are: 12, 8 and 6.
