Centroid (centre of gravity) of Cardboard Quadrilateral

**Instructions**

a) To find the centroid (centre of gravity or balancing point) of a cardboard quadrilateral, divide the quadrilateral by drawing a diagonal and connect the centroids of the two formed triangles with a line - click on the *Step 1* button.

b) Next draw the other diagonal and repeat the process. Clearly, the centroid (balancing point) of the cardboard quadrilateral now has to lie somewhere on the one line, *as well as* on the other. Hence, it has to be at their intersection - click on the *Step 2* button.

**Note**: In physics terms, a 'cardboard' figure is assumed here as being a two-dimensional shape (lamina) of uniform density.

**Experimental Testing**

Cut out a few irregular shaped quadrilaterals from cardboard, then find their balancing points as described above. Next check with a flat tipped pen or sharp pointed eraser that they indeed balance at the constructed points.

**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 when equal masses or weights are placed at the vertices - click on the *Show Point Mass Centroid* button. Also see the dynamic geometry sketch at *Point Mass Centroid of Quadrilateral* for more information.

**Parallogram 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 centroid of a cardboard 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 pentagon, hexagon, etc. by dividing them up into suitable triangles, quadrilaterals, etc.?

3) 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.

**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)

Point Mass Centroid of 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)

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

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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Release: 2020Q2, Semantic Version: 4.6.2, Build Number: 1047, Build Stamp: 139b185f240a/20200428221100

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