Let $(X,d)$ be a uniquely geodesic metric space, i.e. for every $x,y\in X$, there is a unique continuous function $f:I=[0,\ell]\to X$ such that $f(0)=x,f(\ell)=y, d(f(r),f(s))=|r-s|\ \forall r,s\in I,$ where $\ell=d(x,y)$. The image of $f$ is called the geodesic path joining $x,y$. The geodesic midpoint of $x,y$ is $f(\frac{1}{2}\ell)$.
Let $A,B,C$ be three points in $X$. Let $M$ be the geodesic midpoint of $A,B$. My question is: Is it always true that $d(M,C)\le \max\{d(A,C),d(B,C)\}?$