I'm trying to find derivations for the ellipse and the hyperbola that use geometric proofs such as the one found for the parabola on wikipedia (image for clarity)
I'm aware that there are proofs using, Dandelin spheres, but I'm looking for proofs using mostly triangle relations and at most trigonometry, akin to the tools that the ancient greeks had when they initially derived them, but using modern notation (if possible).
So im looking for proofs for the hyperbola and ellipse that
- Are geometric in nature and use elementary triangle relationships, trigonometry or eucludean geometry concepts.
- Do not rely on Dandelin spheres
- Start from a double cone and a plane and end in a form similar or equivalent to the modern standard equations, like $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ for the hyperbola.
I'm kind of at a lost here, I did find a few resources that try to address this issue, but they are either very convoluted or the images are missing/hard to understand. I am about to start reading the actual apollonius conics as translated and commented by Thomas Heath to try to find the answer as well, but I feel this should not be that difficult.