Parabola

Synthetically, a parabola is the locus of points in a plane that are equidistant from a given line (the directrix) and a given point (the focus). Alternatively, a parabola is a conic section obtained as the intersection of a right circular cone with a plane parallel to a generator of the cone.

Let $d$ be a line and $F$ a point. In the special case when $F$ is a point of $D$, the "parabola" with directrix $D$ and focus $F$ is the line through $F$ that is perpendicular to $D$. In the language of conic sections, this corresponds to the case when the plane contains a generator of the cone.

To avoid this degenerate case, assume that $F$ does not lie in $D$, let $\Pi$ be the unique plane containing $F$ and $D$ and let $P$ be the parabola with focus $F$ and directrix $d$. The line $s$ through $F$ and perpendicular to $D$ is called the axis of the the parabola $P$ and is the unique line of symmetry of $P$. The unique point $V$ of $s$ that is equidistant from $F$ and $D$ lies on $P$ and is known as the vertex of the parabola, and the distance $FV$ (half the distance from $F$ to $D$) is called the focal distance of the parabola.

Now let $P'$ be any other parabola in $\Pi$ be with the same focal distance as $P$. Let $F'$ be its focus and $D'$ its directrix. Then there is a unique, orientation-preserving rigid motion of $\Pi$ taking $F$ to $F'$ and $D$ to $D'$ and therefore, the parabola $P$ to the parabola $P'$. In other words, any two parabolas with the same focal distance are congruent.