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.

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.