2
$\begingroup$

Given any triangle $\triangle ABC$, we can always build three ellipses, each of them having foci in two of the vertices and passing through the third one, as shown in the following picture:

enter image description here

In general, these three ellipses intersect in $6$ points $D,E,F,G,H,I$.

Given any two among these $6$ points, it is always possible to build an hyperbole with foci in the given points an passing by one vertex of the triangle $\triangle ABC$.

My conjecture is that

Any pair of these $6$ points represents the two foci of at least one hyperbole passing by at least two of the three vertices of $\triangle ABC$.

enter image description here

In this example, I choose the points $D$ and $F$ as foci. The hyperbole with foci in $D$ and $E$ and passing through $A$ (magenta), pass also through $B$.

Is it possible to prove this conjecture with a compact proof?

Thanks for your help! I apologize in case of mistakes, obscurity or trivialities.

$\endgroup$
2
$\begingroup$

In your picture, since $D, F$ are on the same ellipse with foci $A$ and $B$, so we have $AD+BD = AF+BF$. Hence we have $AF - AD = BD - BF$, which directly implies that the hyperbole with foci $D$ and $F$ and passing by $A$ also passes $B$.

$\endgroup$
  • $\begingroup$ Yes, but this happens also in case the points do not belong to the same ellipse. I added a picture in the question. Thanks anyway!!! $\endgroup$ – user559615 Oct 18 '18 at 17:18
  • $\begingroup$ @AndreaPrunotto In your new picture, both are contained in the red ellipse. You can check that if you choose 2 points among $DEFGHI$, both are contained in one of the three ellipses. $\endgroup$ – Seewoo Lee Oct 18 '18 at 17:48
  • $\begingroup$ Of course! You're right! Thanks again!!! $\endgroup$ – user559615 Oct 18 '18 at 17:58
  • $\begingroup$ I have notice that (still using my picture), the points $G$ and $E$ allow to build a hyperbole passing through all the three vertices of $\triangle ABC$. Any idea why these two points? $\endgroup$ – user559615 Oct 18 '18 at 18:01

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy