What is the length of the marked segment?



The marked segment is the radius of the small circle with center \(C\) .
Call \(r\) its length.
Consider the triangle \(ABC\) that has a right angle at \(B\).
We know that:
\(AB = 1 - r\)
\(BC = \dfrac{1}{2}\)
\(AC = 1 + r\)
By Pythagoras' theorem we have \[(1-r)^2 + \left(\dfrac{1}{2}\right)^2=(1+r)^2 \] which gives \[r = \dfrac{1}{16}\,.\]