Editing Ellipse (section)

== Focus-to-focus reflection property ==
[[File:Ellipse-reflex.svg|250px|thumb|Ellipse: the tangent bisects the supplementary angle of the angle between the lines to the foci.]]
[[File:Elli-norm-tang-n.svg|250px|thumb|Rays from one focus reflect off the ellipse to pass through the other focus.]]
An ellipse possesses the following property:
: The normal at a point <math>P</math> bisects the angle between the lines <math>\overline{PF_1},\, \overline{PF_2}</math>.

;Proof
Because the tangent line is perpendicular to the normal, an equivalent statement is that the tangent is the external angle bisector of the lines to the foci (see diagram).
Let <math>L</math> be the point on the line <math>\overline{PF_2}</math> with distance <math>2a</math> to the focus <math>F_2</math>, where <math>a</math> is the semi-major axis of the ellipse. Let line <math>w</math> be the external angle bisector of the lines <math>\overline{PF_1}</math> and <math>\overline{PF_2}.</math> Take any other point <math>Q</math> on <math>w.</math> By the [[triangle inequality]] and the [[angle bisector theorem]], <math>2a = \left|LF_2\right| < {}</math><math>\left|QF_2\right| + \left|QL\right| = {}</math><math>\left|QF_2\right| + \left|QF_1\right|,</math> so <math>Q</math> must be outside the ellipse. As this is true for every choice of <math>Q,</math> <math>w</math> only intersects the ellipse at the single point <math>P</math> so must be the tangent line.

; Application
The rays from one focus are reflected by the ellipse to the second focus. This property has optical and acoustic applications similar to the reflective property of a parabola (see [[whispering gallery]]).

Additionally, because of the focus-to-focus reflection property of ellipses, if the rays are allowed to continue propagating, reflected rays will eventually align closely with the major axis.