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Consider a closed, simple, smooth curve $C$. Can someone help me prove the following:

For all point in $C$ and their respective neighborhoods, the radius of curvature cannot be infinite.

I basically came up at this point while trying to find some properties of closed curves. But, as one might've understood already that i want to prove that a closed simple plane curve MUST have either a vertex, or a curved region(radius of curvature is finite). Can someone give some hints as to how to prove this, or just disprove this claim?

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  • $\begingroup$ NOTE: this isn't a homework question, and i'm just asking for hints on how to approach this. $\endgroup$ – Lelouch Mar 13 '17 at 8:18
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    $\begingroup$ Perhaps consider the point farthest from the origin. $\endgroup$ – lhf Mar 13 '17 at 9:21
  • $\begingroup$ ok, ill consider that, thanks, any other ideas are also welcome. $\endgroup$ – Lelouch Mar 13 '17 at 9:43
  • $\begingroup$ $C$ might contain a piece of a straight line. Here the radius of curvature is infinite. Are you referring to the four-vertex-theorem? en.wikipedia.org/wiki/Four-vertex_theorem $\endgroup$ – Michael Hoppe Mar 13 '17 at 14:34
  • $\begingroup$ No, i just want to prove that a closed simple plane curve cannot be composed of ONLY finite flat regions, that is it must have a vertex or a finite region with finite radius of curvature. $\endgroup$ – Lelouch Mar 13 '17 at 14:55

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