TITLE: Why ellipse is a "conic" section?
QUESTION [12 upvotes]: When I was first introduced to conic sections, I was kinda surprised that the ellipse is one of them. I mean intuitively, if a cone is cut by a slant surface, one would expect that the cross-section is an egg-shaped one; since the flat circular sections become wider the more we go down.
It is still strange for me that why the cross-section of a cone and a slant surface becomes a symmetric ellipse and not an egg-shaped one. Can anyone explain this in an intuitive way? An answer with a mathematical proof would be nice, but becomes perfect has it been followed by some intuition.

REPLY [2 votes]: I just came across a beautiful illustration in a MO thread and recalled this question. Knowing that this question is quite old, I can't resist to add this one (it was also mentioned in the comments).
The key idea is the fact that tangents to a circle/sphere have equal length. So the only remaining part is to prove that for every cross-section not parallel to the cone's axis, such spheres exist (which is not so hard ;). Then for every $P$ on the cross-section we can write:
$$|F_1P|+|F_2P|=|P_1P_2|$$