Let $\alpha:I\to\Bbb R^3$ be a curve with $|\alpha|=1$ and $\alpha(I)\subset S_{r,c}^2$, where $r>0, c=(c_1,c_2,c_3),$ $S_{r,c}^2=\{(x,y,z)\in\Bbb R^3:(x-c_1)^2+(y-c_2)^2+(z-c_3)^2=r^2\}$ is a sphere. Prove that $\alpha(I)$ is in a maximum circumference $\iff$ lines generate by the normal vector of $\alpha$ intersect in the center $c$.
I've tried to use that two of those lines are
$\lambda N(s_1)=r\alpha ''(s_1)+\alpha(s_1)$ and $\mu N(s_2)=r\alpha ''(s_2)+\alpha(s_2), \lambda, \mu \in \Bbb R$, where N(s) is the normal vector of $\alpha$ in the point $\alpha(s)$, and then made them equal but I don't know how to continue that to obtain that they intersect in the center $c$.