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 points P of 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 of P
from the focus F_{1} to the distance from the line
d is 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.
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