From Wikipedia, the centroid $\mathbf{C}$ of a finite set of points $\mathbf{x}_1, \mathbf{x}_2, \cdots, \mathbf{x}_k$ in $\mathbb{R}^n$ is:
$$
\mathbf{C} = \frac{1}{k} \left( \mathbf{x}_1 + \cdots + \mathbf{x}_k \right)
$$
Is it true that, for convex shapes, $\mathbf{C}$ would also coincide with the intersecting points of lines connecting opposite vertices? Like what I am illustrating in the diagram below.
This statement sounds a bit doubtful for its validity, but I don't really have a solid explanation or arguments to prove/disprove it. Any pointers?
Best Answer
The property you propose doesn't make sense for example for a pentagon. What would be the opposite of a vertex?
Also, you should be aware that the centroid of a polygon is in general distinct from the centroid of its vertices.
For any triangle, the point where its medians concur is both the centroid of the polygon and also the centroid of its vertices.
For parallelograms, the point where the diagonals intersect is again both, the centroid of the parallelogram and the centroid of its vertices.
But in general, for quadrilaterals, the centroid of the quadrilateral, the centroid of its vertices and the intersection of its diagonals are three distinct points.