Can anyone see a simple way of explaining why the area of the triangle ( regardless of where the tangent is drawn) is always the same? In other words why does it work? I can justify it algebraically.
suppose the point of tangency is $x_0$
then the tangent line will have slope $-\dfrac {1}{x_0^2}$ and will include the point $\left(x_0, \dfrac{1}{x_0}\right)$
Given this line see if you can figure out where the x and y intercepts occur.
The area of the triangle is then trivial to compute.
Spoiler:
It is the reciprocal function of $y=x$. This gives it tremendous symmetry across that line. What you are seeing with this area is one manifestation of that symmetry. I am not sure what you are looking for as a "geometric way of seeing the result". But, I would start with that line of symmetry.