I am asked to show that every translation of the euclidean plane can be written as two reflections.
How do I proceed (algebraicly)?
My idea is to proof it in a sense of creating a rectangular triangle which creates the translation vector as the resulting vector of two reflections.
Suppose $A(a, b)$ is the object and $B(a’, b’)$ is its image after a translation (the dotted line).
Locate the (green) line which is $L_1 : x = \frac {a + a’}{2}$
Locate the (blue) line which is $L_2 : y = \frac {b + b’}{2}$
A is first reflected about the $L_1$ to $C$ and then reflected about $L_2$ to $B$.