Investigate
Where is the centroid (balancing point or centre of gravity) of a quadrilateral located if equal weights are placed at the vertices?
Point Mass Centroid (centre of gravity) of Quadrilateral
Theorem
The respective centroids C', D', A' and B' of triangles ABD, ABC, BCD and CDA of quadrilateral ABCD form a quadrilateral A'B'C'D', similar to the original and scale factor -1/3 (a halfturn and reduction by 1/3), with lines AA', BB', CC' and DD' concurrent in G. Then this point of concurrency G (centre of similarity between ABCD and A'B'C'D') is defined as the (point mass or vertex) centroid of the quadrilateral.
Challenge
Can you explain why (prove that) the concurrency result is true?
If stuck, see my paper at Generalizations involving maltitudes.
Experimental Testing
Make a few irregular shaped quadrilaterals from stiff wire, beams or rods and place heavy balls or objects of equal weight at the vertices, then find their balancing points as described above. In order to check that they indeed balance at the constructed points you will have to use thin, much lighter wire connected to the vertices to create a point in the open interior of the quadrilateral in order to balance it there.
Further Generalization
This (homothetic) concurrency result for the point mass or vertex centroid of a quadrilateral generalizes to any polygon as shown in Yaglom (1968), and is applied in another, interesting related generalization at Triangle Centroids of a Hexagon form a Parallelo-Hexagon as well as in Generalizing the Nagel line to Circumscribed Polygons by Analogy.
Other ways of location
In the physical, real world context, this point mass centroid is the centre of gravity or balancing point of equal point masses placed at the vertices of a polygon. Click on the Show Varignon Parallelogram button to see why this is the case for a quadrilateral - for example the masses at the vertices can be replaced by point masses at the midpoints of the sides, which in turn will balance at the centre of the Varignon parallelogram (the intersection of its diagonals), which as you can see above, coincides with the point G.
.....Another way of determining the point mass (vertex) centroid of a quadrilateral (or polygon in general) is to take the average of the coordinates of the vertices. Use the provided tools on the left to check that this is the case.
Note: The centroid of a cardboard quadrilateral (a planar quadrilateral of uniform density), unlike the case for a triangle, does NOT always coincide with the point mass centroid illustrated above - click on the Show Cardboard centroid button. Also see the dynamic geometry sketch at Centroid of Cardboard Quadrilateral.
Parallelogram Centroid Theorem
Regarding the above Note, the following interesting & important theorem holds:
The point mass centroid G and the cardboard centroid of a quadrilateral coincides, if and only if, the quadrilateral is a parallelogram.
Challenge: First try to prove it yourself - before reading the translation from German of a neat proof by Arnold Kirsch in Humenberger (2023). An alternative solution is also provided in Kim et al (2016).
Bisect-diagonal Quadrilateral Theorem
A 'bisect-diagonal' quadrilateral is a quadrilateral with at least one of its diagonals bisecting the other. The following interesting theorem in relation to a bisect-diagonal quadrilateral holds:
If ABCD is a bisect-diagonal quadrilateral with diagonal BD bisected (cut in half) by diagonal AC, then the cardboard (lamina) centroid and point mass centroid both lie on diagonal AC. More-over, if P is the midpoint of AC, then the distance between the cardboard centroid and the point P is twice that of the distance between the cardboard centroid and point mass centroid.
Challenge: First try to prove it yourself - before reading De Villiers (2021).
Explore Further
1) Is the point mass centroid of a quadrilateral always inside? Specifically check by dragging until the quadrilateral becomes concave.
2) Can you find another way of locating the point mass centroid of a quadrilateral with equal masses at the vertices using the coordinates of its vertices?
3) How would you locate the point mass centre in 2) above, if different masses are located at the vertices?
4) Can you figure out different ways of finding the point mass centroid of a pentagon, hexagon, etc. with equal or different masses at the vertices?
5) Where would you locate the balancing point of a 'perimeter' quadrilateral? For example, of a quadrilateral consisting of just sticks or rods forming its perimeter?
References & Readings
De Villiers, M. (2021). Some more properties of the bisect-diagonal quadrilateral. The Mathematical Gazette, Volume 105 , Issue 564 , November, pp. 474 - 480.
De Villiers, M. & Jahnke, H.N. (2024). Using Scientific Principles in Mathematics, pp. 2570-2576, Excerpt from: "The Intimate Interplay Between Experimentation and Deduction: Some Classroom Implications", pp. 2551-2586. In Sriraman, B. (Ed.). Handbook of the History and Philosophy of Mathematical Practice, Springer Nature, Switzerland. DOI: https://doi.org/10.1007/978-3-031-40846-5.
Humenberger, Hans. (2023). Centroids of Quadrilaterals and a Peculiarity of Parallelograms. At Right Angles, November, pp. 1-9.
Hanna, G. & Jahnke, H.N. (2002). Arguments from Physics in Mathematical Proofs: An Educational Perspective. For the Learning of Mathematics, Vol. 22, No. 3 (Nov.), pp. 38-45.
Kim, D-S.; Lee, K.S.; Lee, K.B.; Lee, Y.I.; Son, S; Yang, J.K. & Yoon, D.W. (2016). Centroids and Some Characterizations of Parallelograms. Commun. Korean Math. Soc. 31, No. 3, pp. 637–645. DOI: http://dx.doi.org/10.4134/CKMS.c150165
Oliver, P.N. (2001). Pierre Varignon and the Parallelogram Theorem. Mathematics Teacher, Vol. 94, No. 4, April, pp. 316-319.
Shilgalis, T.W. & Benson, C.T. (2001). Centroid of a Polygon—Three Views. The Mathematics Teacher, Vol. 94, No. 4 (April), pp. 302-307.
Yaglom, I.M. (1968). Geometric Transformations II. Washington, DC: MAA, pp.24; 108-109.
Related Links
The Center of Gravity of a Triangle (Rethinking Proof activity)
Balancing Weights in Geometry as a Method of Discovery & Explanation
Experimentally Finding the Medians and Centroid of a Triangle
Experimentally Finding the Centroid of a Triangle with Different Weights at the Vertices (Ceva)
Centroid of Cardboard (Lamina) Quadrilateral
Centroid (balancing point) of Perimeter Quadrilateral
Three different centroids (balancing points) of a quadrilateral
More Properties of a Bisect-diagonal Quadrilateral
Geometry Loci Doodling of the Centroids of a Cyclic Quadrilateral
Napoleon's theorem (Rethinking Proof activity)
A side trisection triangle concurrency
Generalizations involving maltitudes
Triangle Centroids of a Hexagon form a Parallelo-Hexagon & its Further Generalization
Quadrilateral Balancing Theorem: Another 'Van Aubel-like' theorem
An associated result of the Van Aubel configuration and some generalizations
Van Aubel Vertex Centroid & its Generalization
Generalizations of a theorem of Sylvester (about forces acting on a point in a triangle)
Napoleon's Theorem: Generalizations & Converses
The 120o Rhombus (or Conjoined Twin Equilateral Triangles) Theorem
Jha and Savarn’s generalisation of Napoleon’s theorem
Dao Than Oai’s generalization of Napoleon’s theorem
Euler line proof
Euler-Nagel line analogy
Nine Point Conic and Generalization of Euler Line
A further generalization of the Euler line
Spieker Conic and generalization of Nagel line
Generalizing the Nagel line to Circumscribed Polygons by Analogy
The quasi-Euler line of a quadrilateral and a hexagon
External Links
Centroid (Wikipedia)
Homothety (Wikipedia)
Nine-point circle (Wikipedia)
Euler line (Wikipedia)
Spieker circle (Wikipedia)
Nagel Line (Wolfram MathWorld)
Napoleon's theorem (Wikipedia)
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Created by Michael de Villiers, 6 April 2010; updated 17 October 2021; 16 March 2024; 15 July 2024; 21 Nov 2025.