**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) centroid of the quadrilateral.

Point Mass Centroid (centre of gravity) of Quadrilateral

**Challenge**

Can you explain why (prove that) the result is true?

If stuck, see my paper at *Generalizations involving maltitudes*.

This concurrency (centroid) result 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*.

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 parallelogram (the intersection of its diagonals), which as you can see above, coincides with the point *G*.

**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 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).

**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**

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.

Yaglom, I.M. (1968). Geometric Transformations II. Washington, DC: MAA, pp.24; 108-109.

**Related Links**

Experimentally Finding the Medians and Centroid of a Triangle

Experimentally Finding the Centroid of a Triangle with Different Weights at the Vertices (Ceva)

A side trisection triangle concurrency

Centroid of Cardboard (Lamina) Quadrilateral

Generalizations involving maltitudes

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)

Napoleon's Theorem: Generalizations & Converses

The 120^{o} Rhombus (or Conjoined Twin Equilateral Triangles) Theorem

Jha and Savaran’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

A further generalization of the Euler line

Spieker Conic and generalization of Nagel line

Generalizing the Nagel line to Circumscribed Polygons by Analogy

Euler and Nagel lines for Cyclic and Circumscribed Quadrilaterals

The quasi-Euler line of a quadrilateral and a hexagon

**External Links**

Centroid (Wikipedia)

Nine-point circle

Euler line

Spieker circle

Nagel Line

Napoleon's theorem

**Free Download of Geometer's Sketchpad**

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Created by Michael de Villiers, 6 April 2010; updated 17 October 2021; 16 March 2024.