Editing Parabola (section)

=== 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.