More Properties of a Bisect-diagonal Quadrilateral

NOTE: Please WAIT while the applet below loads.

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.




Back to "Dynamic Geometry Sketches"

Back to "Student Explorations"

Created by Michael de Villiers, 10 August 2020; updated 5 September 2020; 16 March 2024.