I'm not sure if this does what you want, but the subgroup of order $3$ in the additive group $(\mathbb{Z}_{57},+)$ is $\{0,19,38\}.$

To expand on that, one construction for the plane $\mathbb{P}_{7}$ of order $7$ is to take as points the elements of $\mathbb{Z}_{57}$ with lines  $$\ell_k=[k,k+1,k+3,k+13,k+32,k+36,k+43,k+52]$$ for $k \in \mathbb{Z}_{57}.$ Of course the lines are unordered sets but keeping the order might make the cyclic collineation $\phi:x \mapsto x+1$ easier to follow.

The triangle $T=T_0=\{0,19,38\}$ (mentioned above) has $19$ cosets $T_i=\{i,i+19,i+38\}$. 

The lines obtained by extending the sides are

$\ell_6=[6,\ \ 7\ ,9,\ \ {\large 19 },{\large 38},42,49,1]$

$\ell_{25}=[25,26,34,{\large 38 },{\large 0},4,11,20]$

$\ell_{44}=[44,45,47,{\large 0 },{\large 19},23,30,39]$

If "the" center of a triangle $\{P,Q,R\}$ in some plane (like $\mathbb{R}^2$ or $\mathbb{P}_{7}$) is "the" solution $C$ to  $C+C+C=P+Q+R$ then we see that in $\mathbb{P}_{7}$ (as constructed here) a triangle has three centers.  $C,C+19,C+38$ for a unique $0 \leq C \leq 18.$ These are just the vertices of $T_C.$

So the special triangles could be described those which are the triangle of centers for some triangle. OR as those triangles which coincide with their own triangle of centers.