REGULAR POLYGONS AND EULER LINES

Let A1A2A3 be an equilateral triangle and Pa point.
Denote:
1, 2, 3 = the Euler lines of PA1A2,PA2A3, PA3A1, resp.
1,2,3 are concurrent.
Hyacinthos 21592

GENERALIZATION
Let A1A2A3...A3k be a regular 3k-gon and P a point.
Denote:
1, 2, 3,.... 3k = the Euler lines of PA1A2, PA2A3, PA3A4,.....PA3kA1, resp.
These triads are concurrent
1, k+1, 2k+1
2, k+2, 2k+2
3, k+3, 2k+3
4, k+4, 2k+4
......
k, 2k, 3k

k = 2 (Regular Hexagon)
1, 3, 5
2, 4, 6

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