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I had some troubles with this problem :

Let $ABCD$ be a convex quadrilateral. $M$ and $N$ are the midpoints of the diagonals $AC$ and $BD$. The sides $AB$ and $CD$ are extended until they intersect. The intersection point is $E$. The sides $AD$ and $BC$ are extended until they intersect. The intersection point is $F$. Let $P$ be the midpoint of the segment $[EF]$. Prove that $M$, $N$, $P$ are collinear.

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First, I found that a quadrilateral in which the opposite sides interesect is also known as a complete quadrilateral. Then, the line $M-N-P$ is known as Newton-Gauss line and the problem above as Newton's Problem.

I've taught about solving it using areas. I've used the property that median divides the triangle in two echivalent triangles (with the same area). Many properties can be derived from it.

I haven't figure out, but I'm interested in a proof using areas. I would appreciate some suggestions.

Thanks!

Lee David Chung Lin
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scummy
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2 Answers2

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Let $S$ denote the area of $ABCD$.

Lemma 1: The locus of points $X$ in the interior of $ABCD$ such that $|XAB|+|XCD|=\frac{S}{2}$ is a line segment.

In triangle $ADE$, consider a translation of $AB$ to $ES$ and $CD$ to $ET$. Thus, $|XES|=|XAB|$ and $|XET|=|XCD|$. Thus, $|XAB|+|XCD|=|XES|+|XET|=|XSET|=|EST|+|XST|$. Since $S$ and $T$ are fixed points, it follows that $|XST|$ is constant, so the locus of $X$ is the segment made by the intersection of the line parallel to $ST$ with $ABCD$. Further note that if $X$ is outside $ABCD$, then we have $|XAB|-|XCD|=\frac{S}{2}$ or $|XCD|-|XAB|=\frac{S}{2}$.$_\square$

Note that since $M$ is the midpoint of $AC$, $|AMB|=|BMC|$ and $|AMD|=|CMD|$, so $|AMB|+|CMD|=\frac{S}{2}$, so $M$ lies on the locus of $X$. Similarly, $N$ lies on the locus of $X$.

Now note that since $EP=PF$, $|PAB|=\frac{1}{2}|FAB|$ since the distance between $P$ and $AB$ is half of the distance between $F$ and $AB$. Similarly, $|PCD|=\frac{1}{2}|FCD|$. Thus, we have $|PAB|-|PCD|=\frac{1}{2} (|FAB|-|FCD|) = \frac{S}{2}$, so $P$ also lies on this line.

Therefore, $M$, $N$, $P$ are collinear.

Sharky Kesa
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  • What are S and T point mentioned? – r ne Sep 27 '23 at 04:16
  • @rne They are mentioned as part of the construction: translate $AB$ so $A$ maps to $E$, and define the corresponding translated point $B$ maps to as $S$. The definition of $T$ is similar. – Sharky Kesa Oct 14 '23 at 13:46
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In fact there is an alternate, nice, proof using areas that I got, using a link found in the Wikipedia article, in the book by Johnson, "Modern geometry; An Elementary Treatise on the Geometry of the Triangle and the Circle, Houghton Mifflin Ed., 1929" available online (public domain), page 61-62. I reproduce here the two pages.

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Nice, isn't it ?

Jean Marie
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    The reason behind concluding that the midpoints of $AA'$,$AN$ and $AS$ are collinear is : In $\triangle ABC$ , the midpoints of $AB$, $AC$ and cevian $AD$ are collinear. The proof comes from Mid-point theorem. – S Das May 17 '22 at 08:16