The locus of the intersections of the tangents at the end of chords of an ellipse which pass through a common point is a straight line.
A bit of a mouthful, but play with the diagram and it will become clear.
Create a triangle which has 3 points on the circumference of an ellipse, and two sides passing through the foci. Look at 2 curves formed by the third side: the locus of its center and its envelope. One is an ellipse, the other is not.