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Is there synthetic proof of the equation of hyperbola using this definition: A hyperbola is the set of all points in a plane such that the difference of the distances from two fixed points (foci) is constant?

1b3b
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    This appears to be a duplicate of your previous question. Please clarify how what you want here differs from what this very good answer has already provided you. – Blue May 08 '20 at 12:42
  • @Blue, unfortunately that was not entirely synthetic geometry proof. Proof is much much better then traditional algebra proofs, but again, it is not synthetic (part with asymptotes is completely what I was looking for in that question) I couldn't believe that it would be so hard to find an answer to such a natural question as this one :( – 1b3b May 08 '20 at 12:56
  • You should make these clarifications in the question itself; comments are easily overlooked. Since this question has been closed as a duplicate, be as specific as possible about how you intended this question to be different, in order to garner re-open votes. Speaking of comments ... Leave one under @Aretino's answer indicating that the answer is not, in fact, "exactly what [you were] looking for" as you had described; perhaps a few revisions to that answer would suffice to meet your goal. Also, link from the previous answer to this one, so that readers know your quest continues. – Blue May 08 '20 at 13:06
  • At that moment, everything that use geometry was "allowed", let alone such a really good answer. Algebraic manipulations in “ordinary” proof are simply awful, (but useful of course). Man always wants more, complete understanding, and algebra rarely offers that. Thanks for the suggestion, I will keep an eye on that in the future. – 1b3b May 08 '20 at 13:14
  • @1b3b I second Blue's request for clarifications: what exactly would you change in my old answer? Be warned that a little "algebra" in a geometric proof can be easily turned into "completely geometric" passages, at the expense of a much longer (and thus less clear) reasoning. High-school proofs have little algebra because they are extremely easy: as soon as you need more complex proofs, algebra (or something equivalent to it) is necessary, and ancient masters in Geometry (Euclid, Apollonius, Archimedes) largely used it. – Intelligenti pauca May 08 '20 at 16:23
  • @Aretino, your proof is good. But I wonder is there more elegant (faster and with less algebra) proof. Is there answer WHY is equation exatly in that form, etc? Your proof doesn't give me clear picture, or "why". But I said, answer is good for previous answer so I gave you + 1 :) – 1b3b May 18 '20 at 14:08

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