In a triangle, it is well-known and easy to show that the balancing point (centroid, centre of mass) of a 'cardboard' triangle is located at the common intersection (point of concurrency) of the medians. More-over, if equal point masses are place at the vertices of the triangle then the balancing point remains the same.
A free classroom worksheet with guided activity & proof is available in my Rethinking Proof with Sketchpad book (pp. 51-56; 163-168 (teacher notes)), and information is provided inside at the beginning on where to download the accompanying Sketchpad sketches from.
Note: In physics terms, a 'cardboard' figure is assumed here as being a two-dimensional shape (lamina) of uniform density.
Investigate
1) How do we locate the balancing points of respectively:
a) a cardboard (lamina) quadrilateral
b) a quadrilateral with equal masses at the vertices
c) the perimeter of a quadrilateral (ignoring the interior or weights at the vertices)?
2) Do these three different balancing points always coincide? If not, for which kinds of quadrilaterals do they coincide?
Cardboard (Lamina) Quadrilateral
3) Where is the balancing point of a 'cardboard' quadrilateral located?
Think about it a bit before checking your answer at: Centroid of Cardboard (Lamina) Quadrilateral (Easy)
Equal Point Mass Quadrilateral
4) Where is the balancing point of a quadrilateral with equal masses at the vertices located?
Think about it a bit before checking your answer at: Point Mass (Vertex) Centroid of Quadrilateral (Easy)
Perimeter Quadrilateral
If we consider a triangle formed by three sticks (in other words, just its perimeter) its balancing point is located at the incentre of the median triangle (called the Spieker centre) - which is NOT the same as the point of intersection of the medians where both the cardboard and vertex (point mass) centroids of a triangle lie.
5) Where is the balancing point of a perimeter quadrilateral located? (In other words, all the weight is concentrated on the perimeter of the quadrilateral.)
Think about it a bit before checking your answer at: Centroid (balancing point) of Perimeter Quadrilateral (Easy-Medium)
Related Links
Experimentally Finding the Medians and Centroid of a Triangle
Experimentally Finding the Centroid of a Triangle with Different Weights at the Vertices
The Center of Gravity of a Triangle (Rethinking Proof activity - concurrency of medians, Ceva's theorem)
Balancing Weights in Geometry as a Method of Discovery & Explanation
Geometry Loci Doodling of the Centroids of a Cyclic Quadrilateral
External Link
Centroid (Wikipedia)
Spieker centre (Wolfram MathWorld
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Michael de Villiers, 6 April 2010; updated 17 October 2021; 15 March 2023; updated 21 Nov 2025.