Metric/Normed space: midpoints

Question

for a metric space X and points x, y ∈ X we say the point m ∈ X is the midpoint of x and y if $$ d(m,y) = d(m,x) = \frac{1}{2}d(x,y) $$ How to prove that if X is a normed space then $$ m = \frac{x+y}{2}? $$

I tried with $$ d(\frac{x+y}{2}, y) = ||\frac{x+y}{2}-y|| = ||\frac{x-y}{2}|| $$ and this is the same as $$ d(\frac{x+y}{2}, x) $$ but I don't know if this works.

Answer 1

Yes, it works, since $d(x,y)=\|x-y\|$ and therefore both $d(x,m)$ and $d(m,y)$ are equal to $\left\|\frac{x-y}2\right\|$.