How to find the vertices of a multivariable function given D is the boundary?

Question

so here is a function that i am supposed to find its max min given an interval (boundary): $$F(x,y) = xy+2x^2$$ and $D =\{(x,y)\in\Bbb R^2\mid 0 \le y\le x, x\le2\}$.
My problem is I cannot see how this shapes a triangle rather than a square. My professor argues that the boundary tells us we have the following vertices only $$(0,0), (2,0), (2,2)$$ and the thing that i am struggling to understand why isn't $(0,2)$ included in the set of vertices? the question never stated that $D$ is a triangle so it should take any shape it could right?

Answer 1

Hint: $D$ is the intersection of three half-planes (what half-planes?).