Investigate
Where is the centroid (balancing point or centre of gravity) of a 'perimeter' quadrilateral located? (In other words, the centroid of a quadrilateral just consisting of thin rods, beams or stiff wire, all of uniform density, so that all its weight is concentrated on the edges.)
Centroid (balancing point) of Perimeter Quadrilateral
Instructions
1) To find the centroid (centre of gravity or balancing point) of a perimeter quadrilateral such as one made of wire along the edges (all the weight is distributed uniformly along the perimeter), first note that the weight of each side is directly proportional to the length of a side, and which is concentrated at the midpoint of each side. Next find the centroid (balancing point) of the opposite midpoints F and H, and also of opposite midpoints E and G. Using the lever principle of Archimedes, determine the centroid H' of FH by dilating H from centre F with a scale factor of DA/(BC + DA). Similarly determine the centroid G' of EG - click on the 'Step 1' button to show these dilations.
(Note: You can drag measurements or calculation in the sketch if they overlap.)
2) Clearly, the centroid (balancing point) of the perimeter quadrilateral now has to lie somewhere on the line F'H'. Applying again the lever principle of Archimedes, we can now determine the perimeter centroid G" by dilating G' from centre H' with a scale factor of (AB + CD)/(AB + BC + CD + DA) - Use the tools on the left to do these constructions or click on the Step 2 button.
Experimental Testing
3) Make a few irregular shaped quadrilaterals from stiff wire, beams or rods, 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 perimeter in order to balance it there.
Other quadrilateral centroids
4) Click on the 'Link to Comparisom with other centroids' button to navigate to a new sketch which shows a comparison between three different centroids for a quadrilateral.
5) When will all three centroids coincide? Investigate!
(See Kim et al, 2016)
Nagel Line for Perimeter Tangential Quadrilateral
Myakishev (2006) provided an interesting generalization of the Nagel line to a circumscribed/tangential quadrilateral by considering the centroid of its ‘perimeter' (in other words, where all the weight is distributed along the boundary), and constructively defining a Nagel point in relation to it.
Explore Further
1) Is the centroid of a perimeter quadrilateral always inside? Specifically check by dragging until the quadrilateral becomes concave.
2) Can you figure out a way of finding the centroid of a cardboard quadrilateral?
3) Where is the point mass or vertex centroid of a quadrilateral located if equal weights are placed at the vertices?
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.
Humenberger, Hans. (2023). Centroids of Quadrilaterals and a Peculiarity of Parallelograms. At Right Angles, November, pp. 1-9.
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
Myakishev, A. (2006). On two remarkable lines related to a quadrilateral. Forum Geometricorum, Vol. 6, pp. 289-295.
Shilgalis, T.W. & Benson, C.T. (2001). Centroid of a Polygon—Three Views. The Mathematics Teacher, Vol. 94, No. 4 (April), pp. 302-307.
Related Links
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
Experimentally Finding the Medians and Centroid of a Triangle
Experimentally Finding the Centroid of a Triangle with Different Weights at the Vertices (Ceva)
Point Mass (Vertex) Centroid of Quadrilateral
Centroid (centre of gravity) of Cardboard Quadrilateral
Three different centroids (balancing points) of a quadrilateral
Generalizations involving maltitudes of a cyclic quadrilateral
Triangle Centroids of a Hexagon form a Parallelo-Hexagon
More Properties of a Bisect-diagonal Quadrilateral
Quadrilateral Balancing Theorem: Another 'Van Aubel-like' theorem
An associated result of the Van Aubel configuration and some generalizations
Generalizations of a theorem of Sylvester (about forces acting on a point in a triangle)
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
Nine Point Conic and Generalization of Euler Line
Spieker Conic and generalization of Nagel line
Euler-Nagel line analogy
Generalizing the Nagel line to Circumscribed Polygons by Analogy
The quasi-Euler line of a quadrilateral and a hexagon
Geometry Loci Doodling of the Centroids of a Cyclic Quadrilateral
External Links
Centroid (Wikipedia)
Dilation (geometry) (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, with WebSketchpad, 22 Nov 2025.