Because three of the circles are tangent at the same point, their centres lie on a straight line. That being the case, if we draw lines connecting all of the circle centres, we can form two triangles which both use the angle ‘x’: the 90,120,70 triangle and the (R+70),120,R triangle.
We can use the law of cosines which states:
Cos(a) = (B^2+C^2-A^2)/(2BC), (where side A is opposite angle a)
If we do so first for the 90,120,70 triangle, we find that cos(x) = 22/27.
Now if we do the same for the large triangle, we already know cos(x), so:
22/27 = (140R + 19300)/(240R + 16800)
22(240R + 16800) = 27(140R + 19300)
1500R = 151500
R = 101
So therefore the radius of the fourth circle is 101.
