# 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*are equal to the half of the apex angle of the cone, which yields that

^{*}PD*PD*=

*PP'*=

*PF*. Hence,

Each pointPof the conic is equidistant from pointFand lined; and thus, by the definition, the conic is a parabola.