2025 AMC 10A Problems/Problem 22
Solution 1
We can use Descartes' kissing circle theorem, where k_i = 1/r_i and k represents curvature while r represents radius: k_4=k_1+k_2+k_3\pm 2\sqrt{k_1k_2+k_2k_3+k_3k_1}.
We are given the radii are 1, 2, and 3, so plugging in these values we get:
$\textbf{k}=\frac{23/6}$ (Error compiling LaTeX. Unknown error_msg) (note that we removed the negative value for k because a circle can't have negative radius), and since the radius is the reciprocal of 
, we get 
as our answer.
