Points on a Stick

"Two points along a straight stick are randomly selected. The stick is then broken at those two points. Find the probability that the three resulting pieces can be arranged to form a triangle."

This is an exercise (#36 on page 151) from Elementary Statistics, 10th Ed., by Mario Triola of Dutchess Community College.  At the end of the exercise is this parenthetical:  "This is possibly the most difficult exercise in this book."

Hmm, this looks fun.  I'll report back when (if!) I make any progress.