First, I would like to show where I want to go...
... because I find this picture beautiful.
Let draw two circles, the first passing by $B$, $C$, $D$ and the second with center in $A$ and passing by $D$ (i.e. with center on the opposite vertex with respect to the side $BC$).
These two circles determine a point $P$, in correspondence of their (other) intersection.
Let us do the same for the other two sides, obtaining other two points $N$ and $O$.
My conjecture is that the triangle $ONP$ is always equilateral.
Although it is likely and obvious/known result, my question is
Is there an elementary proof for such conjecture?
Thank you for your suggestions!