The quasi-Euler line of a quadrilateral and a hexagon

NOTE: Please WAIT while the applets below load.

1) The quasi-Euler line of a quadrilateral: Given a quadrilateral ABCD, denote by G_a, O_a and H_a the centroid, circumcentre and orthocentre respectively of triangle BCD, and similarly, G_b, O_b, H_b for triangle ACD, G_c, O_c, H_c for triangle ABD, and G_d, O_d, H_d for triangle ABC. Let G = G_aG_c ∩ G_bG_d, O = O_aO_c ∩ O_bO_d, H = H_aH_c ∩ H_bH_d. Then the quasi-orthocentre H, the lamina centroid G, and the quasi-circumcentre O are collinear. Furthermore, HG : GO = 2 : 1.
Reference: Myakishev, A. (2006). On Two Remarkable Lines Related to a Quadrilateral, Forum Geom., 6, 289–295.

Instructions: Drag any of the red vertices A, B. C or D below, and click on the buttons to show/hide.

The quasi-Euler line of a quadrilateral & a hexagon

2) The quasi-Euler line of a hexagon: Subdivide the hexagon ABCDEF into the six quadrilaterals ABCD, BCDE, CDEF, DEFA, EFAB, and FABC, and determine the quasi-circumcentre O, the lamina centroid G and quasi-orthocentre H of each quadrilateral. Then the main diagonals of each of the 3 hexagons formed by these centres are concurrent respectively in O, G and H. More-over, O, G and H are collinear and HG : GO = 2 : 1. (Label the respective hexagons as PQRSTU, P_GQ_GR_GS_GT_GU_G and P_HQ_HR_HS_HT_HU_H).

Instructions
1) Use the 'Link to quasi-euler hexagon' button to navigate to the result formulated above.
2) Drag any of the red vertices A, B. C, D, E, or F below, and click on the buttons to show/hide.

Read my paper Quasi-circumcenters and a Generalization of the Quasi-Euler Line to a Hexagon (pdf, 45 KB, Forum Geometricorum, Vol 14(2014), pp. 233-236).

This interactive link includes more on the quasi-circumcentre of a quadrilateral and its analogue, the quasi-incentre.

Correction Note: It has recently (June 2022) come to my attention that there is an error in my proof of Theorem 4 in my 2014 Forum Geometricorum paper above, where it is incorrectly assumed that the three hexagons concerned are affine equivalent. For a correct proof, and further generalizations to n dimensions and non-Euclidean geometries, the following paper is recommended:
Tabachnikov, S. & Tsukerman, E. (2014). Circumcenter of Mass and generalized Euler line. Discrete & Computational Geometry, 51(4):815–836.
Also read the following associated 2015 note by the same authors: Remarks on the Circumcenter of Mass. Arnold Mathematical Journal.