Context : Work on affine functions
Statement : Let E be an affine space over a field $\mathbb{K}$ , we suppose $Dim(E)\geq 2$
Let $f:E\rightarrow E$ be bijective and we suppose that the image by $f$ of an affine line is an affine line.
Show that if $D_1$ and $D_2$ are two coplanar affine lines , then $f(D_1)$ and $f(D_2$) are coplanar.
What i have started : Let's suppose $D_1$ and $D_2$ are two coplanar affine lines. I started by admitting that either $D_1$ and $D_2$ are parallel , either they are intersecting.
In the case they are intersecting : As $f$ is bijective and in particular injective , we have $$f(D_1 \cap D_2)=f(D_1)\cap f(D_2) $$ which shows that in the case they are intersecting in one point , or if they are intersecting everywhere , it happens to be the same for $f(D_1)$ and $f(D_2)$
But after that , i don't know how to treat the parallel case.
Doing a trial with Peter K hint but i'll probably have to correct (And i may not have understood it well):
Let's suppose $(D_1)$ and $(D_2)$ are parallel now. $(D_1)\vert\vert (D_2)\Leftrightarrow $ if $D_3$ intersect $D_1$ then $D_3$ intersect $D_2$.
Let's take $(D_3)$ that intersect $(D_1)$ (And so $(D_2)$ as we suppose them parallel).
Then $f(D_3)$ intersect $f(D_1)$ by the precedent point , and as $(D_3)$ intersect $(D_2)$, $f(D_3)$ intersect $f(D_2)$
Am i missing something ?