Prove that a metric space is intrinsic Let $(X,d)$ be a general locally compact metric space (in particular not a Riemannian manifold). Suppose we don't know if $(X,d)$ is complete. To prove $(X,d)$ is intrinsic. I have to compute the induced intrinsic metric $\widehat{d}$, defined, for each couple of points $x,y\in X$, as the $\inf$ of the lengths with respect to $d$ of the curves from $x$ to $y$. Then I should show $d=\widehat{d}$. 
But suppose it's hard to compute $\widehat{d}$. Are there ways (or theorems) which could help me prove that $(X,d)$ is intrinsic without computing directly $\widehat{d}$?
This  wikipedia  link contains the definitions which we need in this post
Thank you!  
 A: Non-answer (because the argument requires completeness) : It seems that your “intrinsic metric spaces” are those metric spaces also known as length spaces. This theory begins with a proposition:
Proposition. *A complete metric space $(X,d)$ is a length space if and only if for every $x,y\in X$ and every $\varepsilon>0$, there exists $z\in X$ such that $d(x,z), d(y,z)\leq \frac12 d(x,y)+\varepsilon$.*
The proof constructs continuous curves $c\colon[0;d(x,y)]\to X$ such that $c(0)=x$, $c(d(x,y))=y$ and $\mathop{\rm length}(c)\leq d(x,y)+\varepsilon$ by constructing points $c(s)$ such that $|t-s| d(x,y)\leq d(c(s),c(t))\leq |t-s| (d(x,y)+\varepsilon)$ for every $s$ of the form $a d(x,y)/2^n$, with $1\leq a\leq 2^n-1$, and passing to the limit.
If you know that $X$ is locally compact, then $X$ will be a geodesic space (Cohn–Vossen, generalization of the Hopf–Rinow theorem) : For every $x,y\in X$, there exists a geodesic linking $x$ to $y$: a continuous map $c\colon [0;d(x,y)]\to X$ such that $c(0)=x$, $c(d(x,y))=y$ and $d(c(s),c(t))=|t-s|$ for every $s,t\in[0;d(x,y)]$. 
