Centroid (centre of gravity) of Cardboard Quadrilateral

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

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