Focal 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, $\Delta FAA'$ 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, $FB = BB' \lt BA'.$ 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
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