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In quadrilateral $ABCD$ if $AB$ and $CD$ meet at $E$ and $AD$ and $BC$ at $F$ prove that midpoints of $DB, AC, FE$ are collinear.

My attempt: I connected the midpoints of $AC$ and $BD$ and assumed them to meet $FE$ at some point $Z$.

I tried to prove that $Z$ was mid point of $FE$ using Melenaus theorem but it only got complicated. how do I proceed?

Kenta S
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Albus Dumbledore
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  • Same question was asked by you yesterday https://math.stackexchange.com/questions/3765501/problem-on-cevas-theorem – Martin Hansen Jul 23 '20 at 10:10
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    yes but i had not written my attempt – Albus Dumbledore Jul 23 '20 at 10:36
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    Usually you are expected to edit your old question, instead of posting a new one. – WhatsUp Jul 23 '20 at 10:50
  • Question is well known: completed quadrilateral, resulting line is called the Newton-Gauss line. Proof of the result (as an answer) on Math Stack Exchange using areas here; https://math.stackexchange.com/questions/1401244/collinearity-problem-newton-gauss-line

    If you want to use Menelaus’s Theorem then I've found a proof on that here; http://users.math.uoc.gr/~pamfilos/eGallery/problems/Newton.html

    There's a very good introduction to this entire topic at this level here; http://assets.openstudy.com/updates/attachments/50271a1ae4b086f6f9e12809-waterineyes-1344744709180-ge_g1.pdf

     – Martin Hansen Jul 24 '20 at 06:41

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