Editing Parabola (section)

=== 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.