Investigating Centres of Isosceles Trapezia & Kites coinciding with their Centroids

It is well-known that for a triangle its lamina (cardboard or area) centroid and its vertex centroid (of equal point masses placed at the vertices) always coincide (e.g. see The Center of Gravity of a Triangle), but that is not the case for quadrilaterals; see for example:
Vertex Centroid of Quadrilateral
Centroid of Lamina Quadrilateral

In fact, only for a parallelogram does its vertex centroid and lamina centroid always coincide - see Kim et al (2016) & Humenberger (2023).

Exploration
In the (limited) exploration on this webpage, we will investigate the following interesting coincidences and the characteristics of each case. When will the following centres and centroids coincide?
i) the circumcentre O of an isosceles trapezium with its lamina centroid G_A?
ii) the incentre I of a kite with its lamina centroid G_A?
iii) the incentre I of a kite with its vertex centroid G_V?
Note: For a more complete analysis of these types of questions for arbitrary cyclic and tangential quadrilaterals, respectively, consult Humenberger & De Villiers (2026) and Al-Sharif et al (2009).

Investigating Centres of Cyclic & Tangential Quadrilaterals coinciding with their Centroids

Homothetic relationship between G_V and G_A
Readers may find the following Lemma, about the homothetic relationship between G_V and G_A, useful in analyzing & proving the results given below:
Vertex Centroid of Quadrilateral (click on 'Show Cardboard Centroid G_A')

An isosceles trapezium with G_A = O
1) Drag C until G_A and O coincide.
2) Click the 'Show ∠ DSC' button.
3) What do you notice?
4) Change the shape of the isosceles trapezium by dragging some of the other vertices, then repeat 1)-2).
5) Can you also find a relationship between the length measurements using the calculator tool on the left?
(Hint: Sometimes it is much easier to assume that G_A and O coincide, and then deductively analysing such a situation, than merely 'guessing' and 'testing' relationships between empirical measurements.)
6) Check your conjectured or deductively derived observations above by clicking on the 'Link to iso trap G_A = O Accurate' button to open an accurately constructed sketch.
7) Explore the new sketch. Can you drag ABCD into a rectangle? And into a crossed isosceles trapezium?
(In the latter case, does the relation between the parallel sides & the height still hold?)
Challenge
8) Can you prove your observations in 3)-7)?
Check: Compare your solutions with those given here (did you find a different, easier proof?) and in the references below.

A kite with G_A = I
Click on the 'Link to kite G_A = I Explore' button.
1) Drag any of the vertices until G_V and I coincide.
2) Can you find a relationship between the length measurements using the calculator tool on the left?
(Hint: Sometimes it is much easier to assume that G_A and I coincide, and then deductively analysing such a situation, than merely 'guessing' and 'testing' relationships between empirical measurements.)
3) Check your conjectured or deductively derived observations above by clicking on the 'Link to kite G_A = I Accurate' button to open an accurately constructed sketch.
4) Explore the new sketch by dragging 'X'. Can you drag ABCD into a rhombus? And into a concave kite (dart)?
Challenge
5) Can you prove your observations in 2)-4)?
Check: Compare your solutions with those given in the references below.

A kite with G_V = I
Click on the 'Link to kite G_V = I Explore' button.
1) Drag any of the vertices until G_A and I coincide.
2) Can you find a relationship between the length measurements using the calculator tool on the left?
(Hint: Sometimes it is much easier to assume that G_V and I coincide, and then deductively analysing such a situation, than merely 'guessing' and 'testing' relationships between empirical measurements.)
3) Check your conjectured observations above by clicking on the 'Link to kite G_A = I Accurate' button to open an accurately constructed sketch.
4) Explore the new sketch by dragging 'X'. Can you drag ABCD into a square? And into a concave kite (dart)?
Challenge
5) Can you prove your observations in 2)-4)?
Check: Compare your solutions with those given in the references below.

Further Exploration & Generalization
a) One can ask and explore the same type of questions for the perimeter centroid (balancing point of the edges) for cyclic and tangential quadrilaterals as well as other symmetrical quadrilaterals. But such an exploration is left to the enthusiastic reader.
b) One can also ask and explore the same type of questions for cyclic and tangential pentagons, hexagons, etc. or other symmetrical or non-symmetrical higher order polygons. But such an ambitious exploration is left to the enthusiastic reader.
c) In relation to b), the problem of the respective incentres or circumcentres of tangential or cyclic polygons coinciding with their vertex centroids is also of particular importance in the following two results: Sum of Squared Distances related to Incircles & Circumcircles and Sum of Squared Distances to Vertices of Polygons.

AI Experiment
During the course of 2025, in relation to the first dynamic exploration regarding an isosceles trapezoid, I asked both Google AI and ChatGPT the following question: "When does the centroid of a cyclic quadrilateral coincide with its circumcentre?" Perhaps unsurprisingly, both responded fairly similarly. Read De Villiers (in press) in the references below to see how ChatGPT responded to this question.

References
De Villiers, M. (1999, 2003, 2012). Rethinking Proof with Geometer's Sketchpad (free to download). Key Curriculum Press.
De Villiers, M. (In press). How trustworthy is AI in geometry? Learning & Teaching Mathematics, All rights reserved.
Humenberger, Hans. (2023). Centroids of Quadrilaterals and a Peculiarity of Parallelograms. At Right Angles, November, pp. 1-9.
Humenberger, H. & De Villiers, M. (8 July 2026). Investigating Centres of Cyclic and Tangential Quadrilaterals Coinciding with Centroids. International Journal of Mathematical Education in Science and Technology, DOI: 10.1080/0020739X.2026.2686127.
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.
Al-Sharif, A., Hajja, M. & Krasopoulos, P. T. (2009). Coincidences of Centers of Plane Quadrilaterals. Result. Math. 55, 231-247. DOI: 10.1007/s00025-009-0417-6.

External Links
Centroid (Wikipedia)
When is the centroid of a cyclic quadrilateral also its circumcenter? (Ocean of Math)
Cyclic quadrilateral (Wikipedia)
Tangential quadrilateral (Wikipedia)
Aimssec Lesson Activities (African Institute for Mathematical Sciences Schools Enrichment Centre)
UCT Mathematics Competition Training Material
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.)

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Michael de Villiers, created with WebSketchpad, 6 March 2026; updated 15 March 2026; 28 May 2026; 1 June 2026; 9/15 July 2026.