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* = 3*NG*/(*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

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

*Back to "Dynamic Geometry Sketches"*

*Back to "Student Explorations"*

Michael de Villiers, 27 Oct 2007; modified June 2017 using *WebSketchpad*; updated 16/18 March 2024.