Regular plane curves such that the line they determined is mutually orthogonal to both

Question

Let $\alpha(t)$ and $β(t)$ be two regular plane curves such that the line determined by $\alpha$ and $β$ is mutually orthogonal to $\alpha$ and $β$.

Is it really this simple to prove that the segment of line between $\alpha$ and $β$ at t has constant length? Since that line is mutually orthogonal to $\alpha$ and $β$, then we can write (assuming, without loss of generality, that $\alpha$ and $β$ are reparameterized by arclength):

$\alpha - β = kn_\alpha(t) = k_1n_β(t)$, where $n_\alpha$ and $n_β$ are the unit normal vectors to $\alpha$ and $β$. Then:

$|\alpha - β| = |k| = |k_1|$ .

EDIT: Some minor corrections I should have included earlier

Answer 1

The incredibly obvious eluded me (thanks, Ivo):

$(||\alpha - \beta||^2)' = 2(\alpha' - \beta')\cdot (\alpha - \beta) = 2((\alpha - \beta)\alpha' - (\alpha - \beta)\beta') = 2\cdot0 = 0$ since everything inside the parenthesis is 0 (which follows from the fact that the line they determine is mutually orthogonal to both). So $||\alpha - \beta||^2 = c$, $c \in \mathbb{R}$.