Given any triangle $\triangle ABC$, we build the hyperbole with foci in $A$ and $B$ and passing through $C$. The hyperbole always intersects the side of the triangle that is opposite to the vertex through which it pass in two points $D$ and $E$.
Similarly, we can build other two hyperboles, one with foci in $A$ and $C$ and passing through $B$ (red), and one with foci in $B$ and $C$ and passing through $A$ (green), obtaining other $2$ couples of points $F$, $G$ and $H$, $I$.
My conjecture is that
The $6$ points $D,E,F,G,H,I$ always determine an ellipse.
How can I show this (likely obvious) result with a simple and compact proof?
Thanks for your help, and sorry for the trivial question!
This problem is related to this one.