A point is marked on the face of a regular tetrahedron. Prove that the tetrahedron can be split into four equal сonvex polyhedrons in a way that this point is a vertex of one of them.
This problem is from my group theory course. I think there should be used a group action and Burnside's lemma. But I can't come up with a right set to be acted on. The only group connected with a regular tetrahedron I know is a group of all symmetries, so I think I should use it. Please, give me a hint (not a solution). Thanks. (Also I know how to do it if the point is a middle point of some face. Can I use it?)