# Probability of their's constituting a triangle? [duplicate]

We have a line segments with length $l$ then you choose two random points and cut it from these points so that we have three piece of line segments.

What is the probability that these piece constitute a triangle ?

We need to use triangle inequality but I could not manage it. Thanks.

label the points $0$ to $l$. And call the points $p_1$ and $p_2$ with $p_1<p_2$
notice $p_1<.5$ and $p_2>.5$
now, suppose $p_1$ has already been decided, then the fraction of points that would be suitable $p_2$'s is $p_1$ since we need $|p_1-p_2|<.5$.
So the answer is $\int_0^.5p_1 d p_1$ which is $\frac{1}{4}$