Intersection pattern of planar drawing of $K_{3n+1}$

Question

On observing that any plane drawing of the complete graph $K_4$ can be categorised into two types:

• one 3-cycle surrounding the other vertex;
• two intersecting edges;

I am interested thus on whether in general, we also have for any plane drawing of $K_{3n+1}$, existence of either

• $n$ "triangles" surrounding the other vertex;

or

• $n-1$ "triangles" surrounding the crossing point of the other two edges.

Here, I would suggest only considering the plane drawings (continuous map $f:\Delta^{3n}\subset \mathbb R^{3n+1}\to \mathbb R^2$) satisfying:

• any two vertices do not coincide in $ \mathbb R^2$;

• any 3-cycle is embedded in $ \mathbb R^2$.

My greatest curiosity lies in how one should tackle with this problem, other than exhausting oneself with all possible intersecting curves and monitoring all triangles. Thank you very much for any help!