Nine Point Conic and generalization of Euler Line

Nine Point Conic and Generalization of Euler Line

A famous result attributed to Leonhard Euler (1707 – 1783), is the so-called Euler line: the circumcenter (O), centroid (G) & orthocenter (H) of any triangle ABC are collinear (called the Euler line), GH = 2GO and the midpoint of OH is the center of the nine-point circle (P) - see Coxeter & Greitzer (1967).

Below is a generalization of the Euler line and Nine-point circle to a generalized line and conic.

Nine Point Conic and Generalization of Euler Line
If AD, BE and CF are 3 cevians concurrent at H, then a conic drawn through any 5 of the 3 feet of the cevians, the 3 midpoints of AH, BH and CH, and the 3 midpoints of the sides of ABC, passes through the remaining 4 points. In addition, the center of the conic N, the centroid G of ABC and H are collinear, and HN = 3GN.
This general result is therefore not only a generalization of the famous nine-point circle, but also of the Euler line.

Nine Point Conic and generalization of Euler Line

Investigate
1) First click on the buttons on the left. Then drag points A, B, C, D or E. Also drag D or E onto the respective extensions of BC or CA. Drag these points until the conic becomes a hyperbola or a parabola.
2) Dynamically explore a Further generalization of the Euler line (which only involves similarity).
3) View and interact with the 'dual' of this result, which is a generalization of the famous Spieker circle (and Nagel line) to a six-point conic (and associated Nagel line) at: Spieker Conic and generalization of Nagel line.

Historical Note
The nine-point conic appears to have been first described by Maxime Bocher in 1892 in his paper On a nine-point conic, but the associated generalization of the Euler line is not mentioned.

A Reference & Some Proofs
Coxeter, H.S.M. & Greitzer, S.L. (1967). Geometry Revisited. Washington, DC: The Mathematical Association of America.

a) Read my articles A generalization of the nine-point circle and the Euler line (2005) or The nine-point conic: a rediscovery and proof by computer (2006).
b) Christopher Bradley from the University of Bath, who passed away in 2013, produced the following elegant, concise projective proof of the nine-point conic and associated Euler line in 2010: The Nine-point Conic and a Pair of Parallel Lines.

Related Result
The nine point conic is a special case of the six point bicevian conic, which is attributed to Carnot - click on the 'Link to Generalization' button in the preceding URL to view a dynamic sketch of this 'bicevian conic'.

External Links
Nine-point circle (Wikipedia)
Euler line (Wikipedia)
Spieker circle (Wikipedia)
Nagel point (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.)

***************

Free Download of Geometer's Sketchpad & Associated Learning/Instructional Modules on Various Topics

Back to "Dynamic Geometry Sketches"

Back to "Student Explorations"

Michael de Villiers, 27 Oct 2007; modified June 2017 using WebSketchpad; further modified 6 January 2023; 16 Feb 2026.