Editing Parabola (section)

== Proof of the reflective property ==
[[File:Parabel 2.svg|thumb|right|Reflective property of a parabola]]
The reflective property states that if a parabola can reflect light, then light that enters it travelling parallel to the axis of symmetry is reflected toward the focus. This is derived from [[geometrical optics]], based on the assumption that light travels in rays.

Consider the parabola {{math|1=''y'' = ''x''<sup>2</sup>}}. Since all parabolas are similar, this simple case represents all others.

=== Construction and definitions ===
The point E is an arbitrary point on the parabola. The focus is F, the vertex is A (the origin), and the line {{overline|FA}}  is the axis of symmetry. The line {{overline|EC}} is parallel to the axis of symmetry, intersects the {{mvar|x}} axis at D and intersects the directrix at C.  The point B is the midpoint of the line segment {{overline|FC}}.

=== Deductions ===
The vertex A is equidistant from the focus F and from the directrix. Since C is on the directrix, the {{mvar|y}} coordinates of F and C are equal in absolute value and opposite in sign. B is the midpoint of {{overline|FC}}. Its {{mvar|x}} coordinate is half that of D,  that is, {{math|''x''/2}}. The slope of the line {{overline|BE}} is the quotient of the lengths of {{overline|ED}} and {{overline|BD}}, which is {{math|1={{sfrac|''x''<sup>2</sup>|''x''/2}} = 2''x''}}. But {{math|2''x''}} is also the slope (first derivative) of the parabola at E. Therefore, the line {{overline|BE}} is the tangent to the parabola at E.

The distances {{overline|EF}} and {{overline|EC}} are equal because E is on the parabola, F is the focus and C is on the directrix. Therefore, since B is the midpoint of {{overline|FC}}, triangles △FEB and △CEB are congruent (three sides), which implies that the angles marked {{mvar|α}} are congruent. (The angle above E is vertically opposite angle ∠BEC.) This means that a ray of light that enters the parabola and arrives at E travelling parallel to the axis of symmetry will be reflected by the line {{overline|BE}} so it travels along the line {{overline|EF}}, as shown in red in the diagram (assuming that the lines can somehow reflect light). Since {{overline|BE}} is the tangent to the parabola at E, the same reflection will be done by an infinitesimal arc of the parabola at E. Therefore, light that enters the parabola and arrives at E travelling parallel to the axis of symmetry of the parabola is reflected by the parabola toward its focus.

This conclusion about reflected light applies to all points on the parabola, as is shown on the left side of the diagram. This is the reflective property.

=== Other consequences ===
There are other theorems that can be deduced simply from the above argument.

==== Tangent bisection property ====
The above proof and the accompanying diagram show that the tangent {{overline|BE}} bisects the angle ∠FEC. In other words, the tangent to the parabola at any point bisects the angle between the lines joining the point to the focus and perpendicularly to the directrix.

==== Intersection of a tangent and perpendicular from focus ====
[[File:Parabola-antipodera.gif|thumb|right|200px|Perpendicular from focus to tangent]]
Since triangles △FBE and △CBE are congruent, {{overline|FB}} is perpendicular to the tangent {{overline|BE}}. Since B is on the {{mvar|x}} axis, which is the tangent to the parabola at its vertex, it follows that the point of intersection between any tangent to a parabola and the perpendicular from the focus to that tangent lies on the line that is tangential to the parabola at its vertex. See animated diagram<ref name=ET>{{cite journal |last=Tsukerman |first=Emmanuel |title=On Polygons Admitting a Simson Line as Discrete Analogs of Parabolas |journal=Forum Geometricorum |volume=13 |date=2013 |pages=197–208 | url=http://forumgeom.fau.edu/FG2013volume13/FG201321.pdf}}</ref> and [[pedal curve]].

==== Reflection of light striking the convex side ====
If light travels along the line {{overline|CE}}, it moves parallel to the axis of symmetry and strikes the convex side of the parabola at E. It is clear from the above diagram that this light will be reflected directly away from the focus, along an extension of the segment {{overline|FE}}.

=== Alternative proofs ===
[[File:Parábola y tangente-prueba.svg|thumb|right|200px|Parabola and tangent]]
The above proofs of the reflective and tangent bisection properties use a line of calculus. Here a geometric proof is presented.

In this diagram, F is the focus of the parabola, and T and U lie on its directrix. P is an arbitrary point on the parabola. {{overline|PT}} is perpendicular to the directrix, and the line {{overline|MP}} bisects angle ∠FPT. Q is another point on the parabola, with {{overline|QU}} perpendicular to the directrix. We know that {{overline|FP}}&nbsp;=&nbsp;{{overline|PT}} and {{overline|FQ}}&nbsp;=&nbsp;{{overline|QU}}. Clearly, {{overline|QT}}&nbsp;>&nbsp;{{overline|QU}}, so {{overline|QT}}&nbsp;>&nbsp;{{overline|FQ}}. All points on the bisector {{overline|MP}} are equidistant from F and T, but Q is closer to F than to T. This means that Q is to the left of {{overline|MP}}, that is, on the same side of it as the focus. The same would be true if Q were located anywhere else on the parabola (except at the point P), so the entire parabola, except the point P, is on the focus side of {{overline|MP}}. Therefore, {{overline|MP}} is the tangent to the parabola at P. Since it bisects the angle ∠FPT, this proves the tangent bisection property.

The logic of the last paragraph can be applied to modify the above proof of the reflective property. It effectively proves the line {{overline|BE}} to be the tangent to the parabola at E if the angles {{mvar|α}} are equal. The reflective property follows as shown previously.