since the radius of A is 5 and the radius of C is 10, then the diameter of B must be:
45 - 5 - 10 = 30
so the radius of B is 15.
now the slope of the line connecting the center of A and the center of B is:
m = (y2 - y1)/(x2 - x1) = (12 + 15)/(24 + 12) = 27/36 = 3/4
so notice that if we move 4 units horizontally from the center of A and 3 units up as the slope suggests, we end up with a 3-4-5 triangle. the hypotenuse of this triangle will be 5, which is the exact radius of A. the distance between the center of A and the center of B is 20. so the hypotenuse of the triangle that connects the center of A to the center of B is 20. that is we multiply 5 by 4. since this will be a similar triangle to the first, we can multiply the other 2 numbers by 4. that is we move 4*4 units across and 3*4 units up. that is 16 units across and 12 units up. so the center of B will be given by:
where (x,y) is the center of A
(or we could use: 20/y = 5/3 and 20/x = 5/4 to find the x and y units we move from the center of A)