Given any 2 curves, can we prove the existence of a common normal?

Question

One suggestion would be to say the every two curves will have a distance of closest approach, and this distance will be normal to both of them, but this is either: 1)what we are actually trying to prove 2) inherently wrong in the sense, there maybe 2 curves who don't have a common normal so their distance of closest approach wouldn't be normal to them.

Answer 1

What about orthogonal trajectories?

Each family member is normal to the other family member so they do not have a common normal.

Answer 2

If they have parallel tangents at a point, then they have the same normal at that point.

So you only need to differentiate.

Answer 3

If $\gamma_1$ and $\gamma_2$ are two closed smooth curves in the plane then they have $\geq1$ common normals.

Proof. The function $$f(p,q):=\bigl\{|q-p|\bigm| p\in\gamma_1, \ q\in\gamma_2\bigr\}$$ takes a positive maximum at some point $(p_0,q_0)$ of the compact set $\gamma_1\times\gamma_2$. The line $\ell:=p_0\vee q_0$ then is a common normal to both $\gamma_i$, since otherwise there would be points $p'\in\gamma_1$ and $q'\in\gamma_2$ near $p_0$ and $q_0$ with $|q'-p'|>|q_0-p_0|$.