Is my proof correct? Any advice/comments?
If $F$ is an isometry and $L$ is a line segment then $F(L)$ is a line segment.
Assume $F$ is an isometry. Let Points $P, Q$ be the end-points of $L$ and $X$ be a point between them. Then $distance(P, Q) = distance(P, X) + distance(X, Q) = distance(F(P), F(Q)) = distance(F(P), F(X)) + distance(F(X), F(Q))$
Therefore $F(P), F(Q), F(R)$ are collinear and F(L) is a line.