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I need to show that if lines $L_1$ and $L_2$ are parallel then the reflection $r_{L_1L_2}$ is a translation.

I can draw two lines that are parallel and show how a reflection in both results in a translation, but then I have only shown that the result holds for the two lines I have drawn (not all parallel lines).

My thinking is that I then show that if two lines are parallel then they have the same gradient and therefore the point being reflected will always be translated a long a line perpendicular to the parallel lines. However I don't know how to write this mathematically.

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It is in fact quite easy to show with vectors.

Let's call $M_1$ the reflexion of $M$ according to $L_1$ and $M_2$ the reflexion of $M_1$ according to $L_2$; and $H_1,H_2$ the middles of $MM_1$ and $MM_2$ respectively, which are on $L_1$ and $L_2$ by construction.

$\vec{MM_2}=\vec{MH_1}+\vec{H_1M_1}+\vec{M_1H_2}+\vec{H_2M_2}$

But $\vec{MH_1}=\vec{H_1M_1}$ and $\vec{M_1H_2}=\vec{H_2M_2}$, thus

$\vec{MM_2}=2\vec{H_1M_1}+2\vec{M_1H_2}=2\vec{H_1H_2}$ which is constant.

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