The parabola
Let P be an arbitrary point of the
conic. Consider the sphere G which touches the plane of the
conic and the cone at the point F and at circle k,
respectively. Let the generator through P intersect k at
point P'.
Let the intersection of the planes of circle k and the conic be
line d, and let D and P^{*} be the
projections of P to d and to the plane of circle
k, respectively. The angles P^{*}PP' and
P^{*}PD are equal to the half of the apex angle of the
cone, which yields that PD=PP'=PF. Hence,
Each point P of the conic is
equidistant from point F and line d; and thus, by the
definition, the conic is a parabola.
Back to the previous page
