Make a sketch, first.


Let "a", "b" and "c" be the circles radii, in ascending order.

Then you have these equations


    a + b =  8    (1)

    a + c = 10    (2)

    b + c = 12    (3)


You should find three unknowns  "a", "b" and "c"  from these equations.



For it, add the three equations (1), (2) and (3).  You will get


    2a + 2b + 2c = 8 + 10 + 12 = 30,    


or, after dividing both sides by 2,


    a + b + c = 15.     (4)


Now subtract equation (1) from equation (4).  You will get

    c = 15 - 8 = 7.



Next, subtract equation (2) from equation (4).  You will get

    b = 15 - 10 = 5.



Finally, subtract equation (3) from equation (4).  You will get

    a = 15 - 12 = 3.



The problem is just solved.


ANSWER.  The sides of the triangle are  7, 5 and 3 centimetres, in descending order.


         // Fortunately, the triangle inequalities are satisfied with these numbers;
            so such triangle does really exist (!)