Given any triangle $\triangle ABC$, we can draw two ellipses, one with foci in $A,B$ and passing by $C$, and one with foci in $C,B$ and passing by $A$. We always obtain the points $D,E$, where these two ellipses intersect.
What I find nice and interesting (although it is likely an obvious property, sorry in this case!) is that
The ellipse with foci in $D,B$ and passing by $A$ pass also by $C$, whereas the one with foci in $E,B$ and passing by $C$ pass also by $A$,
as illustrated in the following picture.
To prove this, I tried to use the coordinates, but my calculations are too complicated, and I wonder if there is a more elementary way to show this result.
Thanks for your suggestions, and sorry again if this is too trivial!