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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.

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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.

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  • $\begingroup$ Oh this sounds clever! I need to work through the math to verify.... $\endgroup$ Commented May 13, 2023 at 20:07
  • $\begingroup$ This property is called "the associativity of barycenters". $\endgroup$ Commented May 14, 2023 at 12:23

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