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)

enter image description here

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

  1. Are geometric in nature and use elementary triangle relationships, trigonometry or eucludean geometry concepts.
  2. Do not rely on Dandelin spheres
  3. 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.

  • $\begingroup$ Which explicit relation should be shown? $\endgroup$ – dan_fulea Jul 29 at 11:08
  • $\begingroup$ Starting with a cone cut by a plane, by the use of geometry, arrive to an equation over the plane of the curve equivalent to it's standard form. For example, start with a geometric construction by cutting a cone in such a way that it forms a hyperbola, and by geometric arguments end up demonstrating that the resulting curve can be described by the hyperbola standard equation $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ or any equivalent form of it that can be algebraically manipulated to get the standard equation. Thanks! edited question for clarity. $\endgroup$ – Joaquin Brandan Jul 29 at 18:51
  • $\begingroup$ I took a look at the article you link to ("Deriving the Symptom of the Ellipse") and it looks like what you're after. You could try describing where you get lost (or where an image is missing), and this might be easier to answer than supplying you with an entire proof. $\endgroup$ – brainjam Aug 12 at 17:03
  • $\begingroup$ I'd also suggest that you clarify whether you are looking for references to proofs, If that's the case you should add a "reference-request" tag. $\endgroup$ – brainjam Aug 12 at 17:08
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    $\begingroup$ I don't think this will answer your question, but you might find these two pages from George Salmon of interest, especially the footnote on the right. archive.org/details/bub_gb_q8VUeQNZhCsC/page/n333/mode/2up. I look forward to seeing your solution once you've worked it out. $\endgroup$ – brainjam Aug 12 at 20:42

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