More Properties of a Bisect-diagonal Quadrilateral

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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. Investigate the configuration by dragging A, B, C or D.

More Properties of a Bisect-diagonal Quadrilateral

Summary of properties
1) The point P divides, or equi-partitions, quadrilateral ABCD into four triangles, APB, BPC, CPD and DPA, of equal area.
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 centroid G₂.
5) The diagonal AC is the Newton-Gauss line¹ 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.

Published Paper
An article of mine about these properties has appeared in the Mathematical Gazette, Nov. 2021 issue, and is available at: Some more properties of the bisect-diagonal quadrilateral.

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 at the link given right at the bottom of the aforementioned webpage.

Note: ¹Two 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.

Related Links
A Hierarchical Classification of Quadrilaterals
Definitions and some Properties of Quadrilaterals
Van Aubel's Theorem and some Generalizations
A Van Aubel like property of an Equidiagonal Quadrilateral
A Van Aubel like property of an Orthodiagonal Quadrilateral
Quadrilateral Balancing Theorem: Another 'Van Aubel-like' theorem
Theorem of Gusić & Mladinić
Pitot's Theorem for a tangential or circumscribed quadrilateral
Japanese Circumscribed Quadrilateral Theorem
Perpendicular-Bisectors of Tangential Quadrilateral
Similar Parallelograms: A Generalization of a Golden Rectangle property
Golden Quadrilaterals (Generalizing the concept of a golden rectangle)
Cyclic Kepler Quadrilateral Conjectures
A diagonal property of a Rhombus constructed from a Rectangle
Extangential Quadrilateral
Area Parallelogram Partition Theorem
Area Formula for Quadrilateral in terms of its Diagonals
Crossed Quadrilateral Properties
A generalization of a Parallelogram Theorem to Parallelo-hexagons
Angle Divider Theorem for a Cyclic Quadrilateral
Euler and Nagel lines for Cyclic and Circumscribed Quadrilaterals
Bradley's Theorem for a Circumscribed Quadrilateral

External Links
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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