How do you prove the arithmetic mean of vertices of a polygon lies in itself?

Question

There's an n-sided convex polygon with vertices denoted by $A_1(x_1,y_1),A_2(x_2,y_2)..A_n(x_n,y_n)$. Now we draw a point $P(\frac{\sum x_i}{n},\frac{\sum y_i}{n})$, then how do you show that $P$ must be in the polygon, or is there any counterexample?

Answer 1

Consider the vertex V closest to P. Since the polygon is convex, there must exist a line through V such that all of the vertices are on one side of the line (or on the line), while P is on the other side. This clearly contradicts that P is the arithmetic mean, since you can just rotate the coordinate system such that the y-axis is perpendicular to the line in question. This means that the x-coordinate of P is less than the smallest x-coordinate of a vertex, an impossibility. Drawing a picture will make this more clear.