**This problem is a restatement of [this question](https://math.stackexchange.com/questions/2788189/is-there-a-triangle-which-makes-dense-set-of-angles/2788526#2788526), first announced in MathStackExchange.**


We start with a triangle $T$ in the Euclidean plane and we define $A\_n$ as the set of angles of the $6^n$ triangles obtained from the $n$-th [barycentric subdivision](https://en.wikipedia.org/wiki/Barycentric_subdivision) of $T$. (Angles are *geometric* angles, i.e., the angles of $T$ and its children lie in $[0, \pi]$).


What we can say about set $A=\bigcup\_{n=0}^{\infty}A\_n$? 


Is there a specific predefined triangle for which $A$ is dense on $(0,\pi)$?


If no, does there exist another simple alternative to this iterative procedure, i.e., barycentric subdivision, using other [concurrent lines](https://en.wikipedia.org/wiki/Concurrent_lines) in triangles, which achieves our goal?



Is there a triangle which makes dense set of angles by drawing medians?
The answer to the second question is **yes**, for any non-flat triangle $T$, the set of angles $A$ is dense in $(0, \pi)$. This follows from a stronger result of Barany et al. [Theorem 1, 1]:



> 
> 
> > 
> > **Theorem**. Successive barycentric subdivisions of a non-flat triangle contain
> >  triangles which, to within a similarity, approximate arbitrarily closely any
> >  given triangle.
> > 
> > 
> > 
> 
> 
> 


Here is a nice divulgative [article](http://www.ams.org/publicoutreach/feature-column/fc-2017-06) on the dynamics of iterated barycentric subdivisions. It also has a list of further readings on the topic. 
(*The joy of barycentric subdivision*, by Bill Casselman)- 




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[1] I. Barany, A. Beardon and T. Carne, "Barycentric subdivision of triangles and semigroups of Möbius maps", 1996. [MR1401715](http://www.ams.org/mathscinet-getitem?mr=MR1401715).


