I'm learning about conic sections as defined using focci.
Every derivation I find for the hyperbola get's this definition out of the blue: $b²=c²-a²$
It totally looks like a triangle somewhere, but when I see $a$ and $c$ they are parallel and I cant find how to form a triangle from them.
for example in this definition it's just stated
Let's define $b^2 = c^2 - a^2$ and make the substitution into the equation.
With no explaination.
wikipedia also justs states
Remove the square roots by suitable squarings and use the relation $b^{2}=c^{2}-a^{2}$ to obtain the equation of the hyperbola:
For reference, I'm defining $a,b,c$ as in the following image, where
$b$ is a distance along the $y$ axis
$a$ and $c$ are distances along the $x$ axis (to the vertex and focus respectively)
The question: why is the pythagorean theorem applied here, is there a sneaky triangle that can be defined? how/why does $c$ suddenly become a hipotenuse?