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Has this simple proof appeared in literature? This is essentially the same proof as one where you call the vertices $\vec{a},\vec{b},\vec{c}$, and observe that $\frac{\vec{a}+\vec{b}+\vec{c}}{3}$ lies on each of the three medians.

Consider the triangle $\bigtriangleup ABC$ with the sides $AB,BC,CA$ having midpoints $D,E,F$ respectively.

Imagine that the triangle is split up into infinitesimally wide strips, each of whose long sides are parallel to the side $AB$. The center of mass of each of these strips lie on the median $CD$. Hence, the center of mass of the entire triangle also lies somewhere on the line $CD$.

By an identical argument, the center of mass of the triangle lies somewhere on the median $AE$ as well as somewhere on the median $BF$.

Thus in fact, all three medians $AE,BF,CD$ have a common point, which is the center of mass of the triangle!

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    $\begingroup$ Yes, I saw it for the first time in a Mir edition of popular science, let me see if I can find/remember the title. $\endgroup$ – user551819 May 1 '18 at 19:33
  • $\begingroup$ Well, I couldn't find it and my parents have all the books from when I was a kid. But the book has a drawing of a table with three holes and three strings attached in a common knot, passing through the holes and with weights on the other ends that hand down below the table. $\endgroup$ – user551819 May 1 '18 at 20:24
  • $\begingroup$ How is that related to this argument? $\endgroup$ – Aritro Pathak May 1 '18 at 20:31
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    $\begingroup$ That is the picture in the book. Anyway, don't expect such a well known proof to not have appeared in print. $\endgroup$ – user551819 May 1 '18 at 20:34
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Googling triangle median centroid infinitesimal yields this excerpt from page 69 in the 1917 edition of Applied Mechanics by Alfred P. Poorman (highlighting by Google Books):

enter image description here

The proof doesn't explicitly assert that the medians are concurrent, though it's implied by "Therefore the centroid is on any median". (Applied Mechanics is a free ebook!)

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