Playing with an ellipse I discovered the following properties and am now looking for a nice proof or references.
Let's consider an ellipse with foci $A$ and $B$ and let $C$ be a point on it. Let $l$ be the tangent to the ellipse passing through $C$. Let $C_a$ and $C_b$ be the circles with centers resp. A and B passing through $C$.
(P1) Prove that the other intersection point $F$ of the circles $C_a$ and $C_b$ lies on the ellipse.
Let $D = C_a \cap l$, $E = C_b \cap l$ and $H=DA \cap BE$
(P2) Prove that H lies on the ellipse
Let $I=C_a\cap HF$ and $J=C_b\cap HF$
(P3) Prove that the triples $(I, A, C)$ and $(J, B, C)$ are collinear.
Let $K=FH\cap l$, $L=DI\cap CJ$ and $M=CI\cap EJ$
(P4) Prove that $K$, $L$ and $M$ are collinear (is this some form of Pascal's theorem?)