Reflective properties of parabola

Parabola is a conic defined by its focal property: there is a point - focus - and a line - directrix - and parabola is the locus of points equidistant from the focus and the directrix.

Let A lie on a parabola. Then the tangent to the parabola at A makes equal angles with AF and AA'.

Proof

By definition, FAA' is isosceles. Let T be the midpoint of FA'. Then the perpendicular bisector AT divides the plane into two parts: one consists of points that are nearer to F than they are to A'; the other consists of points that are nearer to A'. Except for A, all points of the parabola lie in the former half. Indeed, let B be a point on the parabola. Then, since BB' is the shortest segment from B to the directrix, FB = BB' < BA'. In particular, B does not belong to AT. We conclude that A is the only point of intersection of that line with the parabola. Therefore, AT is tangent to the parabola at A.

The property is so simple it has been framed as a proof without words, see [Nelsen, p. 44].

References

1. R. Nelsen, Proofs Without Words, MAA, 1993

Copyright © 1996-2008 Alexander Bogomolny