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 Ga, Oa and Ha the centroid, circumcentre and orthocentre respectively of triangle BCD, and similarly, Gb, Ob, Hb for triangle ACD, Gc, Oc, Hc for triangle ABD, and Gd, Od, Hd for triangle ABC. Let G = GaGc ∩ GbGd, O = OaOcObOd, H = HaHcHbHd. 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, PGQGRGSGTGUG and PHQHRHSHTHUH).

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.

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By Michael de Villiers. Created, 23 April 2014; updated 28 July 2020; 18 July 2022; 9 August 2022.