The distinction between $\displaystyle >$ and $\displaystyle \geq$ doesn't matter here.
WLOG let the segments be $\displaystyle a, b, c$ with $\displaystyle a \leq b \leq c$.
Triangle inequality: $\displaystyle a+b > c$
In fact this is equivalent to $\displaystyle c < \frac{1}{2}$, for the given constraints. ($\displaystyle a+b+c=1$ so we can substitute using $\displaystyle a+b=1-c$.)
(Edited out a false start.)
So we have two variables x, y with uniform probability distribution in [0,1]. I made a sketch:
The shaded area is the probability we want, which is $\displaystyle p = \frac{2}{8} = \frac{1}{4}$.