The question is:
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 distance $d$ of $p$ (Fig. 1-36). Let $h$ be the length of the segment determined on $L$ by $C$ and $T$ (thus, $h$ is the "height" of $C$ relative to $T$ ). Prove that
$$
|k(p)|=\lim _{d \rightarrow 0} \frac{2 h}{d^2}
$$
where $k(p)$ is the curvature of $C$ at $p$.
Here is my attempt: 
I consider the osculating circle at p and use the Secant theorem at q which is the intersection point of L and T. Then when $d\rightarrow 0$, we get an equation $$d^2=h_{1}(h_{1}+2r)$$ where $r=\frac{1}{k(p)}$ is the radius of the osculating circle at p. $h_{1}\rightarrow h$ and $h\rightarrow 0$ when $d \rightarrow 0$, so we can deduce that $$d^2=h(\frac{2}{k(p)})$$ and get the result.
But this is not rigorous, can someone help me?