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The question is

Prove that the lines joining the midpoints of opposite sides of a quadrilateral and the line joining the midpoints of the diagonals of the quadrilateral are concurrent.

I have seen other posts on this, and most of them tend to use vectors or coordinate geometry.

For example, this post: Prove that in a quadrilateral, the lines joining the midpoints of the opposite sides and the midpoints of the diagonals are concurrent

I would like to understand this problem in a way using Euclidean Geometry (Lines, Triangles, Circles, etc.). What I have tried is that the quadrilateral formed by the midpoints of opposite sides is a parallelogram. But I can't seem to get much further with this. I understood the analytical geometry part of this, but my study requires this question to be done only by Euclidean Geometry (as I'm preparing for Math Olympiad, and they tend to give extra credits to elegant solutions)

Any help would be appreciated.

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  • $\begingroup$ Please link to the "other posts on this". They may provide insights that help people with the approach you want, without having to duplicate the effort. ... Of course, you should always include your own efforts at a solution, showing what you've tried, where you got stuck, etc. (Add any context and clarification to the body of the question. Comments are easily overlooked and may be hidden.) $\endgroup$
    – Blue
    Jun 4, 2022 at 9:44
  • $\begingroup$ @Blue Sure, have made the required edits $\endgroup$
    – Aarush Mishra
    Jun 4, 2022 at 9:51
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    $\begingroup$ The second answer in the question you linked looks fine to me. $\endgroup$
    – Intelligenti pauca
    Jun 4, 2022 at 11:49

1 Answer 1

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enter image description here

Consider quadrilateral ABCD. As can be seen in figure, quadrilateral MPNQ is a parallelogram because points M, N, P and Q are midpoints of sides of ABCD. So segments PQ and MN bisect each other. In other words, segment MN crosses the midpoint of segment PQ. Also quadrilateral PFQE is parallelogram because $PE||FQ$ and $PE=FQ$, so it's diagonals also bisect each other. That is, EF cross the mid point of PQ; in this way lines MN, PQ and EF are concurrent.

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    $\begingroup$ MPNQ rather than MNPQ. $\endgroup$
    – user376343
    Jun 4, 2022 at 19:30
  • $\begingroup$ Nice, elegant and simple. Thanks! $\endgroup$
    – Aarush Mishra
    Jun 5, 2022 at 15:17

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