hyperbolic (Example 3.16, $\cite{OOS07}$).

\[synchro\] Let $G=G_0* G_1$ with $G_i$’s LHG. Then $G$ is LHG if and only if $G_0$ and $G_1$ are synchronized LH.

Let $G_1,G_2$ are synchronized LH. Let $\Delta=T_1T_2T_3$ be a geodesic triangle in $G:=G_1*G_2$ with geodesic sides $T_1,T_2,T_3$, where $G_i$s are hyperbolic groups with finite presentation $G_i={\langle}X_i|R_i{\rangle}$ for $i=1,2$. Since $\Delta=1$ in $G$, consider the normal form $\Delta=\prod_{i=1}^{k}g_i^{-1}r_i^{\pm 1}g_i$ in the free group $\mathcal{F}(X_1\cup X_2)$ for some positive integer $k$ where $g_i\in (X_1\cup X_2)^{\pm 1}$ and $r_i\in R_1\cup R_2$. We are going to induct on the sum of lengths of the normal form of $\Delta$. Suppose the sum of length of normal form is $\leq 3$. Then the triangle $\Delta$ belongs to one or the free factors either $G_1$ or $G_2$ since the product is $1$. Let us assume that the sum of lengths of normal form is $n\geq 4$. Let we have a cancellation in $T_1T_2=T_1'xyT_2'$ where $xy=1$ in $G_1$ (respectively in $G_2$). Then the triangle $\Delta $ can be written as union of triangle $\Delta_1=xy$ (which is a bigon with non trivial side $x$ and $y$) and the triangle $\Delta_2$ with sides $T_1',T_2'$ and $T_3$. Notice that sum of lengths normal form of $\Delta_1,\Delta_2$ 