How can I proove that a point in the Cartesian plane belongs to the convex hull of a specific polytype in the plane?

Question

Suppose that we have the set $E=\{(0,0),(1,2),(2,1)\}$ and the points of this set, are the vertices of a triangle. How can I prove that the point $(\dfrac{3}{2},\dfrac{3}{2})$ belongs to the convex hull of the polytype generated by $E$ ?

I think if something can prove this, then it is easy to write it as convex combination of the other points of the set $E$, right?

Answer 1

For a large number of points $P_i$ and a point $Q$, it is faster check whether $Q$ is separated from the points $P_i$ by a hyperplane ( i.e. not in c. h.) or not ( i.e. in the c. h.).

Consider the linear problem: Maximize $l(Q)$, with the condition $l(P_i) \le 0$, where $l(P) = l(x_1, \ldots, x_n) = \sum a_i x_i + b$ ( so you are looking for $(a_1, \ldots, a_n, b)$ ( add in the condition $l(P) \le 1$ to guarantee a bounded solution).