As is well known, the centroid G of a triangle ABC divides, or equi-partitions, the triangle into three triangles, AGB, BGC, and CGA, of equal area as shown below. Since G is the centroid or balancing point of a carboard triangle ABC it makes sense that three triangles surrounding have equal area.
(See for example, the activity: The Center of Gravity of a Triangle - click on 'Link to Centroid Properties' & view the proof in the associated Worksheet)
What if?
What about a quadrilateral? Can we find a similar point P for ANY quadrilateral ABCD that divides, or equi-partitions, it into four triangles, APB, BPC, CPD and DPA, of equal area?
If ABCD is a parallelogram, it should be fairly obvious that P should be placed at the intersection of the diagonals, since it is easy to show that the four triangles formed by its diagonals have equal area. Note that the intersection of the diagonals of a cardboard parallelogram is also its balancing point (centroid).
But what about a general quadrilateral? Is it possible for ANY quadrilateral to always find such a point, and if so, where should P be placed? Does the location, if it exists, then also analogously to the triangle case coincide with its cardboard centroid?
Note: Read more about how to find the cardboard (or lamina) centroid of a quadrilateral at: Centroid (balancing point) of Cardboard Quadrilateral
Explore
1) Use the dynamic sketch below to explore the problem by dragging P to see if you can find such a position.
2) Change the shape of ABCD to see for what kind or type of quadrilateral(s) you can find a position for P that satifies the 'equi-partitioning' condition stated above.
Conjecture
3) What conjectures have you made? Ensure that you check your conjectures by sufficient dragging.
4) Formulate your conjectures.
Challenge
5) Can you explain why (prove that) that your conjectures in 4) are true?
Equi-partitioning Point of a Quadrilateral
Checking
6) Click on 'Link to Solution' button to navigate to a new sketch to check your conjectures in 4).
7) Check & compare your proofs in 5), with those in Gilbert et al (1993), Pillay & Pillay (2010) or De Villiers (2021, 474-475).
References
De Villiers, M. (1999, 2003, 2012). Rethinking Proof with Geometer's Sketchpad (free to download). Key Curriculum Press.
De Villiers, M. (2018). A Cyclic Kepler quadrilateral & the Golden Ratio. At Right Angles, March, pp. 91-94.
De Villiers, M. (2021). Some more properties of the bisect-diagonal quadrilateral. Mathematical Gazette, Volume 105 , Issue 564 , November, pp. 474 - 480.
Gilbert, G. T., Krusemeyer, M., & Larson, L. C. (1993). Problem 37. The Wohascum County problem book. Dolciani Mathematical Expositions, 14(10), Washington, DC: Math. Ass. Amer., pp. 68-70.
Humenberger, H. (2018). Balanced areas in quadrilaterals - on the way to Anne’s Theorem. Australian Mathematics Teacher, 74(3), pp. 16-22.
Humenberger, H. (2023). Centroids of Quadrilaterals and a Peculiarity of Parallelograms. At Right Angles, November, pp. 1-9.
Pillay, S. & Pillay, P. (2010). Equipartitioning and Balancing Points of Polygons. Pythagoras, 71 (July), pp. 13-21.
Related Links
The Center of Gravity of a Triangle (Rethinking Proof activity)
Centroid of Cardboard (Lamina) Quadrilateral
More Properties of a Bisect-diagonal Quadrilateral
Cyclic Kepler Quadrilateral
Area Parallelogram Partition Theorem
Area Ratios (Rethinking Proof activity)
Varignon Area (Rethinking Proof activity)
International Mathematical Talent Search (IMTS) Problem Generalized
Area Formula for Quadrilateral in terms of its Diagonals
Feynman's Triangle: Some Generalizations & Variations
Sylvie's Quadrilateral Area Ratio Theorem
A generalization of a Parallelogram Theorem to Parallelo-hexagons
Some Properties of Bicentric Kites
Diagonal Division Ratios in a Quadrilateral
Quadrilateral Balancing Theorem: Another 'Van Aubel-like' theorem
A Hierarchical Classification of Quadrilaterals
External Links
Quadrilateral (Wikipedia)
Newton–Gauss line (Wikipedia)
Aimssec Lesson Activities (African Institute for Mathematical Sciences Schools Enrichment Centre)
Professor Poobhalan Pillay (YouTube videos on SA Mathematics Olympiad Solutions)
SA Mathematics Olympiad Questions and worked solutions for past South African Mathematics Olympiad papers can be found at this link.
(Note, however, that prospective users will need to register and log in to be able to view past papers and solutions.)
UCT Mathematics Competition Training Material
********************************
Back to "Dynamic Geometry Sketches"
Back to "Student Explorations"
Created by Michael de Villiers, 10 August 2020 with WebSketchpad; updated 17 April 2026.