Prove that midpoints of DC,AE,BF are collinear

euclidean-geometrygeometrytriangles

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 ??

Best Answer

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.

enter image description here

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