Given any triangle $\triangle ABC$, let consider two sides, e.g. $AC$ and $BC$.
We draw two ellipses, one with foci in $A,C$ passing by $B$, and the other one with foci in $B,C$ and passing by $A$, obtaining two points $D$ and $E$ where the two ellipses intersect.
Now, we draw two straight lines, one passing by $A$ and $C$ and the other passing $B$, $C$, annotating the intersection points $F,G,H,I,J,K$ with the two ellipses.
My conjecture is that
Given any triangle $\triangle ABC$, the six points $F,H,D,G,I,E$ and the six points $A,J,D,K,B,E$ always determine two conic-sections.
Thanks for any suggestion for a compact proof!