Let $P = \{p_1, p_2, \dots, p_N\} \subseteq \mathbb{R}^2$ be a finite set of points, so that not all of $P$ lie on the same line, and let $$\text{conv}(P) = \left\{ \sum_{i=1}^N \lambda_ip_i \in \mathbb{R}^2 : \sum_{i=1}^N \lambda_i = 1, \lambda_i \geq 0, \lambda_i \in \mathbb{R} \right\}$$ be its convex hull.
How can we show rigorously that this is equal to the intersection of some $k$ closed halfplanes $H_1, H_2, \dots, H_k$, where $H_i = \{ (x,y) \in \mathbb{R}^2 : a_ix + b_iy \geq c_i \}$ for reals $a_i,b_i,c_i$ with $a_i,b_i$ not both $0$?