[Anteas P. Hatzipolakis]:
Let ABC be a triangle and L the Euler line.
Denote:
(Na), (Nb), (Nc) = the NPCs of OBC, OCA, OAB, resp.
A', B', C' = the midpoints of AO, BO, CO, resp.
The parallel to L through A' intersects again (Nb), (Nc) at Ab, Ac, resp.
Denote:
(Na), (Nb), (Nc) = the NPCs of OBC, OCA, OAB, resp.
A', B', C' = the midpoints of AO, BO, CO, resp.
The parallel to L through A' intersects again (Nb), (Nc) at Ab, Ac, resp.
The parallel to L through B' intersects again (Nb), (Na) at Bc, Ba, resp.
The parallel to L through C' intersects again (Nc), (Nb) at Ca, Cb, resp.
Ma, Mb, Mc = the midpoints of AbAc, BcBa, CaCb, resp.
The centroid of MaMbMc lies on the L.
[Peter Moses]:
Hi Antreas,
a^10 b^2+2 a^8 b^4-9 a^6 b^6+5 a^4 b^8+4 a^2 b^10-3 b^12+a^10 c^2-8 a^8 b^2 c^2+10 a^6 b^4 c^2+7 a^4 b^6 c^2-17 a^2 b^8 c^2+7 b^10 c^2+2 a^8 c^4+10 a^6 b^2 c^4-24 a^4 b^4 c^4+13 a^2 b^6 c^4-b^8 c^4-9 a^6 c^6+7 a^4 b^2 c^6+13 a^2 b^4 c^6-6 b^6 c^6+5 a^4 c^8-17 a^2 b^2 c^8-b^4 c^8+4 a^2 c^10+7 b^2 c^10-3 c^12 : :
Ma, Mb, Mc = the midpoints of AbAc, BcBa, CaCb, resp.
The centroid of MaMbMc lies on the L.
[Peter Moses]:
Hi Antreas,
a^10 b^2+2 a^8 b^4-9 a^6 b^6+5 a^4 b^8+4 a^2 b^10-3 b^12+a^10 c^2-8 a^8 b^2 c^2+10 a^6 b^4 c^2+7 a^4 b^6 c^2-17 a^2 b^8 c^2+7 b^10 c^2+2 a^8 c^4+10 a^6 b^2 c^4-24 a^4 b^4 c^4+13 a^2 b^6 c^4-b^8 c^4-9 a^6 c^6+7 a^4 b^2 c^6+13 a^2 b^4 c^6-6 b^6 c^6+5 a^4 c^8-17 a^2 b^2 c^8-b^4 c^8+4 a^2 c^10+7 b^2 c^10-3 c^12 : :
= lies on these lines: {2,3}, {523,23332}, {2452,23291}, {9530,24930}, {11550,16319}
Best regards,
Peter Moses.
Best regards,
Peter Moses.
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