Three semi-circles have their centers as follows: $(0,0), (2, 0), (-4, 0)$ with radii of $6, 4, 2$, respectively. Determine the radius and center coordinates of a circle that is tangent to all three semi-circles.
My attempt:
Let $C_1 = (x_1, y_1) = (0,0), C_2 = (x_2, y_2)= (2, 0), C_3 = (x_3,y_3) = (-4, 0) $ and let $r_1 = 6 , r_2 = 4, r_3 = 2 $.
Further, let $C_4 = (x_4, y_4)$ be the center of the required circle, and $r_4$ be its radius, then by connecting the centers of the fourth circle with each of the first three, we can write the following three quadratic equations:
$ (x_4 - x_1)^2 + (y_4 - y_1)^2 = (r_1 - r_4)^2 $
$ (x_4 - x_2)^2 + (y_4 - y_2)^2 = (r_2 + r_4) ^ 2$
$ (x_4 - x_3) ^ 2 + (y_4 - y_3) ^ 2 = (r_3 + r_4) ^ 2 $
My question is, how to solve these three equations for the unknowns $x_4, y_4, r_4$?