Why are basic feasible solutions the same as vertices geometrically?

The first line on the Wikipedia page for basic feasible solutions reads, In the theory of linear programming, a basic feasible solution (BFS) is, intuitively, a solution with a minimal number of non-zero variables.

I do not understand this, I could always have the origin within a convex polyhedron, but it is not necessary that the origin is a vertex of my polyhedron, is it? What am I getting wrong here?

• This is only intuition. I recommend reading a textbook or lecture notes. Apr 14 '20 at 21:51
• The text I am reading also has similar statements. I must add that they mention this for linear programs in the equational form, if that helps. Apr 14 '20 at 21:53
• A basic feasible solution is one where $n$ linearly independent inequalities are tight, where $n$ is the dimension. The corresponding equations define a 0-dimensional face of the polytope, in other words, a vertex. Apr 14 '20 at 21:54
• You cannot expect to understand what a BFS is without a proper definition. The Wikipedia quote doesn’t include a definition. Can you add your definition of BFS? Apr 14 '20 at 21:55