2-Dimensional geometry

Question

If $t$, $n$, $t'$, $n'$ are the lengths of tangent, normal, sub-tangent & sub-normal at a point $P(x, y)$ on any curve $y = f(x)$ then prove that $$1/(t^2) + 1/(n^2) = 1/(t'n')$$

Answer 1

In terms of the drawing on the Wikipedia page: https://en.wikipedia.org/wiki/Subtangent: $TA:PA=PA:NA$ (similar triangles), thus $t'n'={PA}^2$. So, we are trying to prove:

$$\frac{1}{{PT}^2}+\frac{1}{{PN}^2}=\frac{1}{{PA}^2}=\frac{1}{{PA}^2}\frac{{TN}^2}{{TN}^2}$$

Knowing that $PA\cdot TN=PT\cdot PN=2\cdot\text{area}\triangle PTN$, and by multiplying both sides with ${PT}^2{PN}^2={PA}^2{TN}^2$:

$${PN}^2+{PT}^2={TN}^2$$

which is true (Pythagoras' theorem).