I have a series of inequalities in three variables. I have found an extremal point (x, y, z) with real values.

I now have to prove that this point is indeed extremal - and from this definition:

a point s in a convex subset C is called extremal if
s ∈ [x,y] ⇒ s=x or s=y for every x, y ∈ C

My first thought was to take a linear combination of two points in the subset, and somehow prove thait one of them had to
be the extremal point in question, but this did not really get me anywhere.