A closed regular planar curve $C$ is said to have constant width $μ$ if the distance between any pair of parallel tangent lines to $C$ is always $μ$. If two points on $C$ have parallel tangent lines, call them opposite.
I've been able to show that the opposite point to $\alpha(s)$ is of the form $\beta(s) = \alpha(s) + \lambda(s) T(s) + \mu N(s),$ for some differentiable real function $\lambda$ defined on the same interval as $\alpha$. I want to show that $\lambda = 0$ everywhere. There are questions on the forum about the same problem, but I'm stuck on an earlier step in the proof.
I've struggled many hours (to my surprise), frustratingly, without producing anything of value. Either a hint or full answer would be welcome.
HINT: Use what it means for $\beta(s)$ to be the point opposite to $\alpha(s)$. What does this tell you about $\beta'(s)$? Now calculate.