I am recently reading definitions of Gromov hyperbolicity. I got stuck on a "trivial" question, that is, given a geodesic triangle $\Delta(x,y,z)$ in any metric space $X$ show that $d(x,[y,z])\leq (y\cdot z)_x+insize\Delta$.
My attempt: let $p$ be the point in $[y,z]$ that realizes $d(x,[y,z])$ and let $i_x,i_y,i_z$ be the internal points in line $[y,z],[x,z],[x,y]$ respectively. By triangle inequality, we have $$d(x,p)\leq d(x,z)-d(z,p)=d(x,i_y)+d(i_y,z)-d(z,p)=d(x,i_y)+d(i_x,p)=(y\cdot z)_x+d(i_x,p).$$ Now, my question reduced to showing $d(i_x,p)<insize\Delta$ which I am stuck on for a while. This is on page 410, Metric spaces of non-positive curvature by Bridson and Haefliger.
Terminologies: by internal points of a geodesic triangle, I mean the unique set of three points such that $d(x,y)=a+b,d(y,z)=b+c,d(x,z)=c+a$ where $a=d(x,i_z)=d(x,i_y),b=d(i_z,y)=d(i_x,y),c=d(i_y,z)=d(i_x,z)$ and $insize\Delta$ is the diamater of $\{i_x,i_y,i_z\}$.
I just realized how stupid I was. The point I forget is that $d(x,[y,z])\leq d(x,q)$ for any $q\in[y,z]$. So we have $d(x,[y,z])\leq d(x,i_x)\leq (y\cdot z)_x+d(i_x,i_z)\leq (y\cdot z)_x+insize\Delta$. One can show using triangle inequality that $(y\cdot z)_x\leq d(x,[y,z])$. If $X$ is Gromov hyperbolic, then there exists $\delta>0$ s.t. $d(x,[y,z])-\delta\leq(y\cdot z)_x\leq d(x,[y,z])$, that is, $(y\cdot z)_x$ and $d(x,[y,z])$ are very closed.