Take two overlapping circles. Examine the locus of the center of the circles tangential to both.
It seems to be an ellipse with foci at the centers of the two circles.
Prove it.

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.

The locus of the intersection of tangents at the ends of chords of a parabola through a given fixed point is a straight line.
This is called the polar line of the given point

The polar line is the locus of the intersections of tangent lines at the ends of chords of teh parabola through a fixed point.
Turns out to be conceptually important - not just a curiosity.

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.