More Properties of a Bisect-diagonal Quadrilateral

Definition
A 'bisect-diagonal' quadrilateral is a quadrilateral with at least one of its diagonals bisecting the other. It appears under the name 'bisecting quadrilateral' in my classification of quadrilaterals at: Hierarchical Classification of Quadrilaterals.
Martin Josefsson (2017) explored several interesting properties of this quadrilateral in his Mathematical Gazette paper at Properties of bisect-diagonal quadrilaterals.
Investigate
The following properties also hold for a bisect-diagonal quadrilateral ABCD as shown below with M the midpoint of the bisected diagonal BD and P the midpoint of diagonal AC (De Villiers, 2021). Investigate the configuration by dragging A, B, C or D.

More Properties of a Bisect-diagonal Quadrilateral

Summary of properties
1) The midpoint P of AC divides, or equi-partitions, quadrilateral ABCD into four triangles, ABP, BCP, CDP and DAP, of equal area (See Gilbert et al; 1993; Pillay & Pillay, 2010)
(Also note that for a quadrilateral to have an 'equi-partitioning' point like this is a necessary and sufficient condition for it to be a bisect-diagonal quadrilateral).
2) The respective centroids E, F, G and H of triangles DPA, APB, BPC and CPD of a bisect-diagonal quadrilateral ABCD form a parallelogram EFGH and the intersection of its diagonals, G₁, lies on AC, and is the lamina centroid of ABCD.
3) The lamina parallelogram EFGH of a bisect-diagonal quadrilateral ABCD is homothetic to the Varignon parallelogram IJKL formed by the midpoints of the sides of ABCD, with the centre of similarity between the two located at P, and a scale factor of 2/3.
4) The distance between the lamina centroid G₁ and the equi-partitioning point P of a bisect-diagonal quadrilateral is twice that of the distance between its lamina centroid G₁ and point mass (vertex) centroid G₂.
5) The diagonal AC is the Newton-Gauss line of the complete bisect-diagonal quadrilateral ABCDQR and therefore passes through the midpoint S of the third diagonal QR. Furthermore, diagonal QR is parallel to diagonal BD.

Notes
a) Two different proofs for the collinearity of the midpoints of the three diagonals of a complete quadrilateral, which define the Newton-Gauss line, are available at: Theorem of Complete Quadrangle. Alternatively, see The complete quadrilateral and its properties.
b) Note that the result about the equi-partitioning point P of a quadrilateral lying on the diagonal bisecting the other one (in other words, lying on the Newton-Gauss line) is a special case of a more general result called Anne's theorem. Let's use the notation (ABC) to represent the are of triangle ABC. In terms of the diagram above for a general quadrilateral ABCD, Anne's theorem states that the locus of the points I so that (ABP) + (CDP) = (BCP) + (DAP) is the line connecting the midpoints of the diagonals (i.e. the Newton-Gauss line) - see Humenberger, 2018). In the special case of the bisect-diagonal quad ABCD, the areas of the four triangles are equal, so obviously the relationship (ABP) + (CDP) = (BCP) + (DAP) holds.

Some Special Cases
6) Apart from parallelograms and kites as obvious special cases of a bisect-diagonal quadrilateral, any cyclic quadrilateral ABCD with its sides AB : BC : CD : DA in geometric progression with common ratio r, is also a bisect-diagonal quadrilateral. For more info, go to Cyclic Kepler Quadrilateral, and also read the associated paper (De Villiers, 2018) at the link given right at the bottom of the aforementioned webpage.
7) As pointed out by Humenberger (2023), the lamina centroid and point mass (vertex) centroid of a bisect-diagonal quadrilatera will only coincide when it is a parallelogram.

References
De Villiers, M. (2021). Some more properties of the bisect-diagonal quadrilateral. Mathematical Gazette, Volume 105 , Issue 564 , November, pp. 474 - 480.
De Villiers, M. (2018). A Cyclic Kepler quadrilateral & the Golden Ratio. At Right Angles, March, pp. 91-94.
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.
Josefsson, M. (2017). Properties of bisect-diagonal quadrilaterals. Mathematical Gazette, 101(551) (July), pp. 214-226.
Pillay, S. & Pillay, P. (2010). Equipartitioning and Balancing Points of Polygons. Pythagoras, 71 (July), pp. 13-21.

Related Links
A Hierarchical Classification of Quadrilaterals
Definitions and some Properties of Quadrilaterals
Cyclic Kepler Quadrilateral
The Center of Gravity of a Triangle (Rethinking Proof activity - concurrency of medians)
Centroid of Cardboard (Lamina) Quadrilateral
Point Mass (Vertex) Centroid of Quadrilateral
Balancing Weights in Geometry as a Method of Discovery & Explanation
Area Parallelogram Partition Theorem
Area Formula for Quadrilateral in terms of its Diagonals
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
The Equi-partitioning Point of a Quadrilateral

External Links
Quadrilateral (Wikipedia)
Newton–Gauss line (Wikipedia)
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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