The tangents $PR$ and $PS$ from an external given point $P$ are equally inclined to the focal distances $PF_1$ and $PF_2$ (where $F_1$, $F_2$ are the foci of the ellipse). (see here, property 1).
I was wondering if the converse is also true:
Given four points $A,B,C,D$, such that points $C$,$D$ lie in the same semiplane determined by $AB$, let $I$ be the intersection of the external bisectors of angles $\angle ADB$ and $\angle ACB$. If $\angle CIA \equiv \angle DIB$, show that $C,D$ lie on an ellipse with foci $A,B$.
Multiple drawings in Geogebra suggest that this is true. However, i could not find a proof. Is this property real? If yes, how can we go about proving it?