Ellipse as Envelope
Suppose we have two points F and G on the diameter of a circle, equidistant from the center O. Given any point P on the circle, the line perpendicular to FP at P is tangent to the ellipse with foci F and G and auxiliary circle O. This allows us to construct an ellipse as an "envelope" of tangents.
In the applet below, the auxiliary circle and foci are shown in white. The rays from the focus are shown in green, and the tangents are shown in yellow. The number of tangents constructed varies from 0 to 180.