In triangle $ABC$, $AD,BE,CF$ are concurrent lines. $P,Q,R$ are points on $EF,FD,DE$ such that $DP,EQ$ and $FR$ are concurrent. Prove that $AP,BQ,CR$ are also concurrent.
I tried to apply Ceva's theorem, but since the endpoints of the cevians are in separate triangles, I got nowhere.
This is true due to the Cevian's Nest Theorem. You can check the proof of it here.