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 quadrilateral
b) and one with equal masses at the vertices?
2) Do these two different balancing points always coincide as they do in the case of a triangle? If not, for which kinds of quadrilaterals do they coincide?
Cardboard Quadrilateral
3) Where is the balancing point of a 'cardboard' quadrilateral located?
Think about it a bit before going 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 going checking your answer at: Point Mass Centroid of Quadrilateral (Medium-Hard)
Third Balancing Point
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.
Similarly, we can investigate where the balancing point of a 'perimeter' quadrilateral is located, and in due course I may add another dynamic geometry page with information in that regard. In the mean time, more information is available through a Google search online.
Related Links
Experimentally Finding the Medians and Centroid of a Triangle
Experimentally Finding the Centroid of a Triangle with Different Weights at the Vertices
External Link
Centroid (Wikipedia)
Free Download of Geometer's Sketchpad
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Michael de Villiers, 6 April 2010; updated 17 October 2021; 15 March 2023.