VMO 1974 Problem 4

$C$ is a cube side $1$ . The $12$ lines containing the sides of the cube meet at plane $p$ in $12$ points. What can you say about the $12$ points?

Dataset text AI-MO AoPS. Quoted with source.

Solutions

No solution is on olymmath yet. Official and human solutions come first when they arrive. AI solutions come after them, with a label.