prove that midpoints of DC,AE,BF are collinear

Question

Question -

a straight line meets AB,BC and AC produced of ABC at D,E,F respectively.prove that midpoints of DC,AE,BF are collinear....

My try -

I suppose that P and Q are midpoints of AE and DC respectively...then I draw lots of parallel lines from them to the sides to use midpoint theorem ...but none of them works. menelaus didn't work either...

Any hints ??

Answer 1

This is the Newton-Gauss line. Knowing the name, you can Google for proofs on the web, including this one at Wikipedia. In case the link goes bad, here's a screen cap.