Here is a cute geometry problem I saw some time ago. I know the solution, I just wanted to share ;-) (Please, don't be mad at me.)
Consider an acute triangle $\triangle ABC$. Let $AP$, $AQ$ and $BP$,$BQ$ be the angle trisectors as shown on the picture below. Prove that $|\angle APQ| = |\angle QPB|$.
Edit: There does exist one very simple and elegant solution, so don't be stumbled if you happen to guess/derive it.