On the plane $\mathbb{R}^2$, I want to show that any translation can be written as a composition of two half-turns $H_a, H_b$ with either $a$ or $b$ chosen arbitrarily.
I know that any composition of two half-turns would give us some translation. But, I am stuck showing the other direction.
First, I picked an arbitrary $x$ and made a translation $T_x$.
Then, I picked an arbitrary $a$ and made a half-turn $H_a$ with this, which is a rotation about $a$ through $\pi$.
But, I am not sure how $b$ should be chosen to make $T_x = H_a H_b$.
I understand that this $b$ must be dependent on fixed $a,x$, but I wonder how.
Fix a vector $v$ and a point $p$; you want to prove that if $H_b$ is a half-turn around some point $b$, then there is some $a\in\Bbb R^2$ such that$$H_a\bigl(H_b(p)\bigr)=p+v\tag1.$$Let $p'=H_b(p)$ and let $a$ be $\frac12\bigl(p'+(p+v)\bigr)$. Then $H_a(p')=p+v$; in other words, $(1)$ holds. So, since $H_a\circ H_b$ is a translation, it has to be the trianslation by the vector $v$.