I was considering the maximum number of points in a 3D-figure such that all the internal line segments of the figure (all the lines that have endpoints as vertices of the figure, and go through the interior of the figure) intersect at that point.
I'm fairly sure the answer is either 0 (it's just not possible) or 1 (it is possible), but how can I rigorously prove this?
Assuming the 3D-figure is a polygon of N vertices, there will be at most $M=N(N-3)/2$ interior line segments. Now, let's assume that all the $M$ line segments pass through two distinct points for now. Since two distinct points uniquely define a line, all lines passing through the same two distinct points must be the same line. Therefore, all the M line segments must be the same line, which is impossible. Therefore, the assumption that all M line segments pass through two distinct points cannot be true and the maximum number of such points can only be either 0 or 1.
Note: This is just a logical deduction. Whether it is a rigorous proof, I will let all true mathematicians be the judge. :-)