Two random breaks giving triangle sides

A one-metre stick is broken at two points chosen independently and uniformly along its length, producing three pieces. What is the probability that these three pieces can be the sides of a triangle?

$\tfrac14$. The pieces form a triangle exactly when every piece is shorter than half the stick. With breaks $x<y$ this means $x<\tfrac12$, $y>\tfrac12$ and $y-x<\tfrac12$, a triangle of area $\tfrac18$ inside the region $x<y$ of area $\tfrac12$.