The cartesian coordinates of the centroid of a set of points in the plane is the mean of their cartesian coordinates.
Is there a geometric way of finding the centroid of an arbitrarily large set of points? Meaning, for example, that given the points plotted on a sheet of paper and a ruler and compass, can we draw lines that will reveal the centroid? Or a tight bound on the location of the centroid?
In the case of three points: we know that the centroid of a triangle is the point of intersection of its medians, which are the lines joining each vertex with the midpoint of the opposite side.
I am wondering if there is some trick that applies to sets of more than 3 points. I am thinking there must be a geometric way of computing the mean coordinates, but also suspecting that this problem has been addressed before.
Here is a simple but not necessarily efficient way.
Take an arbitrarily chosen pair of points. Replace them with a single point at their mid-point and label this point with a $2$.
Consider this new point and another arbitrarily chosen point. Divide the line joining them into 3 equal parts. Replace both points with a point at the dividing position closer to the point with label $2$. Label this new point with a $3$.
Etc.
NOTE If the number of points is even then you can start by simply pairing them up and replacing each pair of points with their midpoint.