



 
Andrew, For a plane to intersect the cone in the points of a straight line, it MUST contain the vertex of the cone. The plane can intersect the cone in a pair of lines, or touch it in a single line. Since all lines must contain the vertex, the pair of lines will necessarily intersect. To be consistent, we must consider the case of the single line to be counted twice  each of its points is a double point; in other words, this does give a pair of parallel lines, but in the degenerate form where the two lines coincide. Chris  


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