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:
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 $\triangle ABC$.
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.
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 $B$.