TITLE: Given a regular compact surface $S$ in $\mathbb{R}^3$ proof there exists a line in $\mathbb{R}^3$ which intersects perpendicularly with $S$ twice
QUESTION [8 upvotes]: I need to proof that given a regular compact  surface $S$ in $\mathbb{R}^3$ there exists a line in  $\mathbb{R}^3$ which intersects perpendicularly with $S$ twice.
Could you help me?

REPLY [12 votes]: Nice question! If you fix $x\in S$ and find the point $y\in S$ farthest from $x$, you can check easily that the chord joining $x$ and $y$ must be normal to $S$ at $y$. By symmetry, then, if we take any pair of points $x,y\in S$ with maximal distance (which exists by compactness), the chord joining $x$ and $y$ must be normal to $S$ at both points.