Definition 1. Line is tangent to the ellipse in the point iff it is the axis of the exterior angle formed by the focal lines of the tangent point .
Figure 1: Tangent to ellipse |
Remark. Tangent line (or tangent, for short) to the ellipse contains no interior points of the ellipse.
See animation 4.
Theorem 1. All intersection points of tangents to the ellipse and their perpendiculars through the ellipse foci and are located on the circle with the centre in the centre of the ellipse and radius equal to the ellipse semimajor axis. Circle is called the auxhiliary circle.
Figure 2: Auxhiliary circle |
See Animation 5.
Theorem 2. All points symmetric to one focus of the ellipse with respect to all tangents to the ellipse are located on the circle with the centre in the other focus and radius equal to the doubled semimajor axis of the ellipse. Circle is called the control circle related to the focus .
Figure 3: Control circle |
See animation 6.
Remark. The control circle related to the focus has the centre in the focus and the same radius equal to the double semimajor axis of the ellipse, .
Theorem 3. There exist two different tangent lines to the ellipse that are passing through an arbitrary exterior point of the ellipse, or that are parallel to an arbitrary direction.
Corollary 1. Tangent points of two tangent lines to the ellipse that are parallel to each other form line segment with the endpoints on the ellipse and midpoint in the centre of the ellipse.