When a metric space is a normed space? I'm trying to figure out that which condition should be provided for a metric space to be normed also?

When the metric is induced from a norm.

This kind of metric space $(X,d)$ must satisfy $$ d(x+a,y+a)=d(x,y)$$ $$ d(\alpha x,\alpha y)=|\alpha|d(x,y)$$ for all $x,y,a\in X$,and scalar $\alpha$.

And $X$ must be a vector space.