I've been trying to solve the following problem:
Let $C$ be a plane curve and let $T$ be the tangent line at a point $p \in C$. Draw a line $L$ parallel to the normal line at $p$ and at a distance $d$ of $p$. let $h$ be the length of the segment determined on $L$ by $C$ and $T$. Prove that $\lvert k(p) \rvert = \lim_{d \to 0} \frac{2h}{d^2}$ where $k(p)$ is the curvature of $C$ at $p$.