Euler-Nagel line analogy (duality)
The following interesting analogy (duality) was already noted in 1916 by Coolidge (1971, reprint):
1) The circumcenter (O), centroid (G) & orthocenter (H) of any triangle ABC are collinear (Euler line), GH = 2GO and the midpoint of OH is the center of the nine-point circle (P).
2) The incenter (I), centroid (G) & Nagel point (N) of any triangle are collinear (Nagel line), GN = 2GI and the midpoint of IN is the center of the Spieker circle (S).
Euler-Nagel line analogy
Comments
i) Note that for the Euler line, a halfturn with centre G and a scale factor of 1/2, maps ABC onto the median triangle A'B'C', and circumcentre O to P. But a dilation with a scale factor of 2 from centre O, maps A'B'C' to A"B"C", and P to H. Therefore, H (the orthocentre of ABC) is the circumcentre of A"B"C".
ii) Similarly, for the Nagel line, a halfturn with centre G and a scale factor of 1/2, maps ABC onto the median triangle A'B'C', and incentre I to S. But a dilation with a scale factor of 2 from centre I, maps A'B'C' to A"B"C", and S to N. Therefore, N (the Nagel point of ABC) is the incentre of A"B"C".
iii) Since the Euler line generalizes to cyclic polygons, a Nagel line for circumscribed polygons can now be generalized by the above analogy as shown by the dynamic sketch at Nagel line for circumscribed quadrilateral.
Some References
Coolidge, J.L. (1971). A Treatise on the Circle and the Sphere. Chelsea Publishing Company, Bronx, NY, pp. 53-57, (original 1916).
De Villiers, M. (2008). Generalizing the Nagel line to Circumscribed Polygons by Analogy and Constructive Defining. Pythagoras, 68 (Dec.), pp. 32-40.
Hofstadter, D.R. (1992). From Euler to Ulam: Discovery and Dissection of a Geometric Gem. Indiana University, Bloomington, Indiana 47408.
Kondratieva, M. (2013). Geometrical Constructions in Dynamic and Interactive Mathematics Learning Environment. Mevlana International Journal of Education (MIJE), Vol. 3(3), pp. 50- 63.
Related Links
Euler line proof
Water Supply II: Three Towns (Rethinking Proof activity, concurrency perpendicular bisectors)
The Center of Gravity (Centroid) of a Triangle (Rethinking Proof activity, concurrency medians, Ceva)
Triangle Altitudes (Rethinking Proof activity, concurrency, orthocentre)
Nine Point Conic and Generalization of Euler Line
Further Euler line generalization
Six Point Cevian Circle & Conic
Spieker Conic and generalization of Nagel line
Point Mass Centroid of a quadrilateral
Euler and Nagel lines for Cyclic and Circumscribed Quadrilaterals
Nine-point centre (anticentre or Euler centre) & Maltitudes of Cyclic Quadrilateral
Generalizing the Nagel line to Circumscribed Polygons by Analogy
The quasi-circumcentre and quasi-incentre of a quadrilateral
The quasi-Euler line of a quadrilateral and a hexagon
Concurrency and Euler line locus result
Collinear Conjecture
Four Collinear Points
Some Trapezoid (Trapezium) Explorations (see Investigations 3 & 4)
An interesting collinearity
Investigating incentres of some iterated triangles
Investigating circumcentres of iterated median triangles
Quadrilateral Similar Triangles on Sides Collinearity
Twin Circles for a Van Aubel configuration involving Similar Parallelograms
External Links
Nine-point circle (Wikipedia)
Euler line (Wikipedia)
Spieker circle (Wikipedia)
Nagel point (Wikipedia)
Theodor Spieker Biography (Wikipedia)
Nagel Line (Wolfram Math World)
The Euler Line and the 9-Point Circle (Cut The Knot)
Aimssec Lesson Activities (African Institute for Mathematical Sciences Schools Enrichment Centre)
UCT Mathematics Competition Training Material
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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Michael de Villiers, 6 April 2010 with JavaSketchpad; converted to WebSketchpad, 17 October 2021; updated 8 April 2025; 16 Feb 2026; 1 Oct 2026.