I'm going through a step-by-step proof that all isometries fixing the origin are linear maps.
Suppose $f$ is an isometry that fixes the origin, and that $u$ and $v$ are any points in $\mathbb R^n$.
(1) Prove that $f$ preserves straight lines and midpoints of line segments.
(2) Using the fact that $u+v$ is the midpoint of the line joining $2u$ and $2v$, and step (2), show that $f(u+v)= f(u)+f(v)$.
It is the second step I'm struggling with. If $f$ preserves midpoints then we have, for $a$, $b \in \mathbb R^n$ that
$$f\left(\frac{a+b}{2}\right) = \frac{f(a)+f(b)}{2}.$$
Substituting $2u$ for $a$ and $2v$ for $b$, we have
$$f\left( \frac{2u+2v}{2}\right)= f(u+v) = \frac{f(2u)+f(2v)}{2},$$
and I just can't get the RHS to simplify to anything reasonable. Obviously we want it to equal $f(u)+f(v)$, but I can't see how to make it happen. I'm also thinking we must begin with the given definition of isometry (i.e. the usual $|f(u)-f(v)|=|u-v|$) but I've tried fiddling around with this definition as well, and can't get anything useful.
Can anyone point out to me what I'm doing wrong, and sorry if I'm missing something obvious.
Since $f$ preserves the origin and midpoints of line segments, $\frac12f(2u)$, which is the midpoint between the origin and $f(2u)$, is also the image of the midpoint between the origin and $2u$, i.e. $u$.