Without knowing the equation for an ellipse, but rather just with the geometric definition i.e. the ellipse is the locus of points with $PF_1 + PF_2 = k > F_1F_2$.
I tried to use a classic construction: $\omega = \odot(F_1,k)$, so for $Q \in \omega$, the point $P$ of meeting of the perpendicular bisector between $Q$ and $F_2$, with $QF_1$ is in the ellipse.
By the converse: given $P$ in the ellipse, the point $Q = PF_1 \cap \odot(P,PF_2)$, not in between $P$ and $F_1$, is in $\omega$.
The problem is that this last operation transforms lines in weird curves, while I was hoping it would transform lines in another lines and thus, a line would meet a circle in more than two points.
Notice that it is not obvious from this definition that an ellipse can be transformed in a circle by dilation on some axis.