# The ellipse

Let *P* be an arbitrary point of the conic
section. Consider the two spheres which touch the cone and the
plane. Let sphere *G*_{1} and *G*_{2} touch
the cone at circles *k*_{1} and *k*_{2}, and
the plane at points *F*_{1} and *F*_{2},
respectively. Let the generator through *P* intersect
*k*_{1} and *k*_{2} at *P*_{1}
and *P*_{2}, respectively. As
*PP*_{1}=*PF*_{1} and
*PP*_{2}=*PF*_{2},
*PF*_{1}+*PF*_{2}=*P*_{1}*P*_{2}.
Since the line segment *P*_{1}*P*_{2} is
bounded by circles *k*_{1} and *k*_{2}, its
length does not depend on the choice of point *P*. Hence,

For all pointsPof the conic,PF_{1}+PF_{2}is constant; and thus, by the definition, the conic is an ellipse.

We can use the spheres to prove another important
property of the ellipse. Let the intersection of the planes of circle
*k*_{1} and the ellipse be line *d*, and let
*D* and *P*^{*} be the projections of *P* to
*d* and to the plane of of circle *k*_{1},
respectively. It is easy to see that the triangles
*PP*^{*}*P*_{1} are similar for all choices
of *P*, thus the ratio of distances *PP*^{*} and
*PP*_{1}=*PF*_{1} is constant. The triangles
*PP*^{*}*D* are also similar, which yields that the
ratio of *PP*^{*} and *PD* is another
constant. Putting the results together, we obtain that

The ratio of the distance ofPfrom the focusF_{1}to the distance from the linedis constant.

The constant is less than 1, because angle
*P*^{*}*PP*_{1} is the half of the apex
angle of the cone, and angle *P*^{*}*PD* is greater.