There a exercise of diferential geometry thats sugests that the point where all the normal lines went throught is the center, the resolution assuming this works.
There is another exercise that wants to prove that if all tangents lines are equidistant from the same point them the curve, $\alpha$, is a circle or a segment of line.
I tried to solve by the equation:
$<\alpha(s)-a,N(s)>=d$, where $a$ is the equidistant point and $d$ is the constant distance from the tangent vector, so by derivating the equation and using the frenet equation and assuming that $a$ is the center i get to $0=0$.
Is my guess correct or that aproach to solving is wrong? my idea was to derivate this equation and use frenet formulas to show that the curvature is null or a non-null constant.
This approach is correct. But you don't get to $0=0$. You should get to $\langle \alpha(s)-a, -\kappa(s)T(s)\rangle = 0$. How does this show the curve is either a (segment of a) line or a circle?