<s>[INST] What is the radius of the black circle tangent to all three of these circles? The red, blue, and green circles have diameters 3, 4, and 5, respectively.
What is the radius of the black circle tangent to all three of these circles?
I just figured out the radius is exactly $\dfrac{72}{23}$ but I don't know how to do the solution.

 [/INST] It is simply the Apollonius Problem and solved by the following three quadratic equations using WolframAlpha for center $ (x,y) $ and radius $r$ of the biggest circle, which is tangent to other three circles;
(1) $ x^2 + (y - \frac{3}{2})^2 = (r - \frac{3}{2})^2 $, circle on side of length $ = 3 $,
(2) $ (x - 2)^2 + y ^ 2 = (r - 2)^2 $, circle on side of length $ = 4 $,
(3) $ (x - 2)^2 + (y - \frac{3}{2})^2 = (r - \frac{5}{2})^2 $, circle on side of length $ = 5 $.
The solution is $ x = \frac{36}{23}, y = \frac{24}{23} $ and radius $ r $ is the Answer $ = \frac{72}{23} $.
</s>