parabola (4,5); (-1,2) parabola (1,4); at (0.15,0.75)[$P^1_n=o_{\omega}(d_n)$]{}; at (5.85,0.75)[$P^2_n=o_{\omega}(d_n)$]{}; at (3,4.75)[$P^3_n=o_{\omega}(d_n)$]{}; (4.90,3.5) parabola (3,0); at (4.25,0.5)[$\leq \alpha_n$]{}; at (0,5)[$\mathbf{H_n}$]{}; at (1,0)[$a_n^{\prime}$]{}; at (5,0)[$b_n$]{}; at (7,2)[$b_n^{\prime}$]{}; at (-1,2)[$a_n$]{}; at (4,5)[$c_n$]{}; at (1,4)[$c_n^{\prime}$]{}; at (3,-0.25)[${A_n}$]{}; at (5,4)[${B_n}$]{}; at (0,3)[${C_n}$]{};

to prove that it contains no simple non-trivial limit geodesic triangles[^1].

Let $\triangle_{\infty}$ be a simple non-trivial limit geodesic triangle in $Con^{\omega}(X,e,d)$ with sides A, B and C with,

$\lim^{\omega}{A_n}=A, 
        \lim^{\omega}{B_n}=B, 
        \lim^{\omega}{C_n}=C$

where $A_n,B_n,C_n$ are geodesics in $X$ with endpoints $a_n',b_n$ for $A_n$ and $b_n',c_n$ for $B_n$ and $c_n',a_n$ for $C_n$, for all $n\in \mathbb{N}$. Let ${\triangle}_{\infty}$ is approached by hexagons $H_n$, formed by vertices $a_n{a_n}'b_n{b_n}'c_n{c_n}'$ in $X$ with $\ell (P^i_n)$[^2]$=o_{\omega}(d_n)$ for $i=1,2,3$, where, $ P^1_n$ is a geodesic joining $a_n,a_n'$ in $X$ and respectively $P^2_n$ is a geodesic joining $b_n,b_n'$ in $X$ and $P^3_n$ is a geodesic joining $c_n,c_n'$ in $X$.

Denote the perimeter of the hexagon $H_n$ by $|H_n|$. Let $A$ be a nontrivial side of $\triangle_{\infty}$. By lemma $\color{blue}{\ref*{hyperbolicgeodesicngon1}}$, $A_n$ belongs to closed $4f_X(|H_n|)$ neighborhood of other $5$ sides in $H_n$. In particular $A_n$ is contained in closed $\alpha_n$ neighborhood of $B_n\cup C_n $, where $$\begin{aligned}
\label{A} 
    \alpha_n=4f_X(|H_n|) + \max_{i=1,2,3}\{\ell(P^i_n)  \}.
    \end{aligned}$$ We have, $$\begin{aligned}
\label{B}
    \ell(P^i_n)=o_{\omega}(d_n)\\
    \ell(A_n) = \Theta_{\omega}(d_n)\nonumber
    