Is the centroid of any triangle can be calculated by averaging vertices

Question

Is it always true that the centroid of any triangle can be calculated by averaging the x and y coordinates of its vertices without bothering with finding medians?

Answer 1

Correct. The centroid of a finite set of points $(x_i,y_i)$, $i=1,\dots,n$ has coordinates $$\bar x = \frac{1}{n}(x_1+\dots+x_n),\quad \bar y = \frac{1}{n}(y_1+\dots+y_n)$$ Similar in higher dimensions. One can also imagine the points being of unequal masses $m_1,\dots,m_n$; then the center of mass has coordinates $$\bar x = \frac{1}{M}(m_1x_1+\dots+m_nx_n),\quad \bar y = \frac{1}{M}(m_1y_1+\dots+m_1y_n)$$ where $M=m_1+\dots+m_n$ is the total mass.