HYACINTHOS 26360

[Antreas P. Hatzipolakis]:

 

 

Let ABC be a triangle.

Denote:

Ab,Ac = the orthogonal projections of A on IB, IC, resp.

La = the Euler line of AAbAc

Laa, Lab, Lac = the reflections of La in IA,IB,IC, resp.

A1B1C1 = the triangle bounded by Laa, Lab, Lac

L1= the Euler line of A1B1C1. Similarly L2 ,L3.


1. L1, L2, L3 are concurrent.

2. the reflections of L1, L2 ,L3 in IA, IB,IC, resp. are concurrent.


[Peter Moses]:

Hi Antreas,

1).
a^2 (a^8 b^3+a^7 b^4-3 a^6 b^5-3 a^5 b^6+3 a^4 b^7+3 a^3 b^8-a^2 b^9-a b^10-2 a^9 b c+a^8 b^2 c+2 a^7 b^3 c-2 a^6 b^4 c+6 a^5 b^5 c-10 a^3 b^7 c+2 a^2 b^8 c+4 a b^9 c-b^10 c+a^8 b c^2-4 a^7 b^2 c^2+a^6 b^3 c^2+5 a^5 b^4 c^2-3 a^4 b^5 c^2+3 a^3 b^6 c^2-4 a b^8 c^2+b^9 c^2+a^8 c^3+2 a^7 b c^3+a^6 b^2 c^3-14 a^5 b^3 c^3+2 a^4 b^4 c^3+10 a^3 b^5 c^3-5 a^2 b^6 c^3+2 a b^7 c^3+b^8 c^3+a^7 c^4-2 a^6 b c^4+5 a^5 b^2 c^4+2 a^4 b^3 c^4-14 a^3 b^4 c^4+4 a^2 b^5 c^4+5 a b^6 c^4-3 b^7 c^4-3 a^6 c^5+6 a^5 b c^5-3 a^4 b^2 c^5+10 a^3 b^3 c^5+4 a^2 b^4 c^5-12 a b^5 c^5+2 b^6 c^5-3 a^5 c^6+3 a^3 b^2 c^6-5 a^2 b^3 c^6+5 a b^4 c^6+2 b^5 c^6+3 a^4 c^7-10 a^3 b c^7+2 a b^3 c^7-3 b^4 c^7+3 a^3 c^8+2 a^2 b c^8-4 a b^2 c^8+b^3 c^8-a^2 c^9+4 a b c^9+b^2 c^9-a c^10-b c^10):: 

on lines {{104, 5663}, {109, 3028}, {513, 3109}, {6789, 10176}}.
reflection of X(3109) in the OI line.

2).
2 a^7-2 a^5 b^2-a^4 b^3-3 a^3 b^4+a^2 b^5+3 a b^6+a^4 b^2 c-b^6 c-2 a^5 c^2+a^4 b c^2+8 a^3 b^2 c^2-a^2 b^3 c^2-3 a b^4 c^2-b^5 c^2-a^4 c^3-a^2 b^2 c^3+2 b^4 c^3-3 a^3 c^4-3 a b^2 c^4+2 b^3 c^4+a^2 c^5-b^2 c^5+3 a c^6-b c^6:: 

on lines {{1, 523}, {30, 944}, {265, 952}, {405, 2452}, {6741, 11735}}.
midpoint X(6742) and X(7984).
reflection of X(i) in X(j) for these {i,j}: {{3109, 1}, {6741, 11735}].

Best regards,
Peter Moses.