## Nine Point Conic and generalization of Euler Line
If *AD*, *BE* and *FG* are 3 cevians concurrent at *F*, then a conic drawn through any 5 of the 3 feet of the cevians, the 3 midpoints of *AF*, *BF* and *CF*, 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 *O* of *ABC* and *F* are collinear, and *FN* = 3*NO*. (On the left, scroll down to see the measurement 'o' = *FN/NO* at the bottom).
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1) Drag points *A*, *B*, *C*, *D* or *E*. Also drag *D* onto the extension of *BC*. What do you notice?
2) Dynamically explore a *Further generalization of the Euler line* (which does not include the nine-point conic).
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)*.
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.
Affine Proof: Christopher Bradley from the University of Bath, who passed away in 2013, produced the following elegant, concise proof of the nine-point conic and associated Euler line in 2010: *The Nine-point Conic and a Pair of Parallel Lines*.
Michael de Villiers, 27 Oct 2007, Created with GeoGebra |