At the webpage Nine Point Conic and Generalization of Euler Line it is shown how the famous nine point circle and its associated Euler line can be generalized further to a nine point conic and a generalization of the Euler line directly related to this conic.
Further Generalization
The preceding 'conic associated' generalization of the Euler line can be further generalized as follows:
Given any triangle ABC with midpoints of the sides X, Y and Z and three cevians concurrent in H as shown below. With H as centre of similarity and scale factor (1/k), construct triangle LJK similar to ABC. Let N be the centre of similarity between LJK and the median triangle XYZ. Then H, N and G are collinear, and HN = 3NG/(k - 1).
(Note: the theorem is now no longer about conics any more, but only about the similarities between the triangles concerned.)
Further Euler line generalization
Explore
1) Click on the buttons in sequence in the sketch to see the construction step by step.
2) Drag any of the red points/vertices to explore.
3) Click on the 'Show 6-point conic' button to show the 6-point conic which passes through D, X, Y, E, Z and F, but note that unlike the Nine Point Conic mentioned at the top, this conic does not necessarily always pass through the points L, J and K.
4) Can you drag L so that these 3 points also (approximately) lie on the conic? What do you notice about the value of k then?
5) Can you drag the figure until you get the 9-point circle (and its associated Euler line)?
6) Explore & prove the special case of the result at Concurrency and Euler line locus result.
Published paper
Read my 2005 article A Generalization of the Nine-point circle and Euler line in Pythagoras.
Application
Challenge: Can you use the same type of similarity of triangles above to further generalize the Nagel line generalization at Spieker Conic and generalization of Nagel line to maintain the collinearity of relevant points and the ratio of segments?
Related Links
Euler line proof
Nine Point Conic and Generalization of Euler Line
Six Point Cevian Circle & Conic
Spieker Conic and generalization of Nagel line
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-Euler line of a quadrilateral and a hexagon
Concurrency and Euler line locus result
External Links
Nine-point circle
Euler line
Spieker circle
Nagel Line
The Euler Line and the 9-Point Circle
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Michael de Villiers, 27 Oct 2007; modified June 2017 using WebSketchpad; updated 16/18 March 2024.