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From the image given below, I want to prove that there exists a unique plane $p \neq P$ s.t. $p \cap$ inclined cone $=$ circle centered at $O_{2}$. I also want to prove that if ray $SO_{1}$ (where $O_{1}$ is the center of the other circle) meets the plane $p$ not in the center of circle $O_{2}$, then $O_{2}$ is not in $SO_{1}$. The help would be appreciated. I am not knowledgeable on planar geometry but if I had to take an educated guess, I would say for the first problem that suppose the planes did not intersect, and then arrive at a contradiction.

Nick Freeman
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  • Every right circular cone can be intersected by a plane orthogonal to its axis in order to obtain a circle. On the other hand, if $O_2$ is a given point outside the given cone, you'll never manage to have it as the center of a circular intersection. The second question appears to be trivially true: “If ray $r$ meets plane $p$ not in $O$ then $O$ (which is on $p$) is not in $r$”. That follows from the fact that a ray intersects a plane in one unique point, so either that point is $O$ or it is not $O$. Did I miss something here? – MvG Oct 23 '14 at 21:52
  • Notice the cone is not a right circular cone. Its a slanted cone. – Nick Freeman Oct 24 '14 at 01:15

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