Παρασκευή 17 Ιανουαρίου 2025

ENNEADIC LINES

Definition
Let T1, T2, T3 be three triangles with the property: The Nine Point Circle centers N1, N2, N3 of T1, T2, T3, resp. are collinear.
I call the line N1N2N3 enneadic line of T1, T2, T3
See an example of an enneadic line in geometry of triangle Euclid 6930

Conjecture
Let A1A2A3... An be a regular n-gon, n>4.  
There are three triangles of the n-gon having an enneadic line.

Problem
How many "different" enneadic lines are in a n-gon ?
(two enneadic lines are "different" if the 2 triads of the triangles are not symmetric)

Similarly we may ask for enneadic lines of 4 triangles.

PICTURES


Pentagon


Hexagon


Heptagon 1


Heptagon 2


Heptakaidecagon


Heptagon (4 triangles. Elias Hagos)

Mail Antreas P. Hatzipolakis

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