1) Euler line
For a cyclic quadrilateral, a dilation of -1/3 with centre G, maps ABCD onto the centroid quadrilateral A'B'C'D', and circumcentre O to O'. But a dilation with a scale factor of 3 from centre O, maps A'B'C'D' to A"B"C"D", and O" to H. Therefore, if H, the circumcentre of A"B"C"D", is defined as the 'orthocentre' of a cyclic quadrilateral, we have HG = 3GO (as well as HP = PO, where P is the nine-point (or Euler) centre).
2) Nagel line
For the circumscribed quadrilateral, a dilation of -1/3 with centre G, maps ABCD onto the centroid quadrilateral A'B'C'D', and incentre I to I', and because of the half-turn, I, G and I' are collinear. Next apply a dilation with a scale factor of 3 from centre I, to map A'B'C'D' to A"B"C"D", and I" to the point N, which we now constructively define as the Nagel point of a circumscribed quadrilateral. Then from the applied transformations we have similarly to the cyclic case that N, G and I are collinear, and NG = 3GI. If we analogously define the Spieker centre S as the midpoint of NI, we also have SG = GI.
Euler and Nagel lines for Cyclic and Circumscribed Quadrilaterals
Further Generalization
Both results respectively generalize to cyclic and circumscribed polygons. For more information read my 2008 Pythagoras paper Generalizing the Nagel line to Circumscribed Polygons by Analogy and Constructive Defining.
Note
Myakishev (2006) provides a completely different generalization of the Nagel line of a circumscribed quadrilateral by considering instead the centroid of its ‘perimeter' (in other words, where all the weight is distributed along the boundary), and constructively defining a different Nagel point.
Reference
Myakishev, A. (2006). On two remarkable lines related to a quadrilateral. Forum Geometricorum, Vol. 6, pp. 289-295.
Related Links
Euler-Nagel line analogy
Point Mass (Vertex) Centroid of Quadrilateral
Centroid (balancing point) of Perimeter Quadrilateral
Three different centroids (balancing points) of a quadrilateral
Generalizations involving the maltitudes of a cyclic quadrilateral
Euler line proof
Nine Point Conic and Generalization of Euler Line
Spieker Conic and generalization of Nagel line
Triangle Centroids of a Hexagon form a Parallelo-Hexagon & its Further Generalization
The quasi-Euler line of a quadrilateral and a hexagon
External Links
Centroid (Wikipedia)
Nine-point circle (Wikipedia)
Euler line (Wikipedia)
Spieker circle (Wikipedia)
Nagel Line (Wolfram MathWorld)
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Michael de Villiers, 6 April 2010; updated to WebSketchpad, 17 October 2021; updated 22 Nov 2025.