Summary: The tangent to an ellipse at a point $P$ can be constructed as the exterior angle bisector of the angle that the point $P$ makes with the two foci of the ellipse. Typically this fact is proved by showing that the exterior angle bisector cannot meet the ellipse in a second point, and therefore must be the tangent. This note gives an alternative proof that is more in line with the notion of tangent as a limit of secants.