Bijection that preserve lines must be linear

Question

There have been some past posts on this topic, but with no complete answer provided. Namely, if T is a bijection of the Euclidean plane that maps line segments to line segments (setwise) then T is an affine transformation, in the sense of linear algebra. What is the most elementary proof of this fact?

Answer 1

Three points $u$, $v$ and $w$ in the plane are collinear iff the union of the line segments $[u, v]$, $[u, w]$ and $[v, w]$ is itself a line segment. So a bijection from the plane to itself that maps line segments to line segments preserves collinearity. See Bijection $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ preserves collinearity $\iff \ \ f(x)=Ax+b$ for the rest of the argument.