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Antreas P. Hatzipolakis
[APH]:
Nab, Nac = the NPC centers of AB'I, AC'I, resp.
Nbc, Nba = the NPC centers of BC'I, BA'I, resp.
Nca, Ncb = the NPC centers of CA'I,CB'I, resp.
Na, Nb, Nc = the NPC centers of IB'C', IC'A', IA'B', resp.
1 The Euler lines of OaOabOac, ObObcOba, OcOcaOcb are concurrent (parallels)
2. The Euler lines of NaNabNac, NbNbcNba, NcNcaNcb are concurrent.
**** 1 The Euler lines of OaOabOac, ObObcOba, OcOcaOcb are parallel with infinity point X(517)
**** 2. The Euler lines of NaNabNac, NbNbcNba, NcNcaNcb are concurrent in
(2 a^9 (b+c)+b c (b^2-c^2)^4-2 a^8 (b^2+c^2)+a (b-c)^4 (b+c)^3 (b^2+b c+c^2)-a^7 (7 b^3+b^2 c+b c^2+7 c^3)+2 a^2 (b^2-c^2)^2 (b^4-2 b^3 c-2 b c^3+c^4)+a^6 (6 b^4-2 b^3 c+8 b^2 c^2-2 b c^3+6 c^4)-a^3 (b-c)^2 (5 b^5+7 b^4 c+6 b^3 c^2+6 b^2 c^3+7 b c^4+5 c^5)+a^5 (9 b^5-4 b^4 c+b^3 c^2+b^2 c^3-4 b c^4+9 c^5)-a^4 (6 b^6-5 b^5 c+2 b^4 c^2+6 b^3 c^3+2 b^2 c^4-5 b c^5+6 c^6): ... : ... ).
The ETC-search numbers: (2.86418637501589, 3.22715415794361, 0.0845486610160832)
On lines {11,500},{30,1319},{496,5495},{511,6713},{549,4271},{952,5453}
**** 3. The parallels through A', B', C' to Euler lines of NaNabNac, NbNbcNba, NcNcaNcb, resp. are are concurrent in
(4 a^8 b c-2 a^9 (b+c)+b c (b^2-c^2)^4-12 a^6 b c (b^2+c^2)-6 a^2 b c (b^2-c^2)^2 (b^2+c^2)-a (b-c)^4 (b+c)^3 (b^2+3 b c+c^2)+a^7 (7 b^3+5 b^2 c+5 b c^2+7 c^3)+5 a^3 (b-c)^2 (b^5+3 b^4 c+2 b^3 c^2+2 b^2 c^3+3 b c^4+c^5)-a^5 (9 b^5+6 b^4 c-5 b^3 c^2-5 b^2 c^3+6 b c^4+9 c^5)-a^4 (-13 b^5 c+6 b^3 c^3-13 b c^5): ... : ... ).
The ETC-search numbers: (-0.0821492377402210, -1.14776402559181, 4.47318537858189)
On lines {11,8143}, {115,119}, {952,5492}, {1317,2771}
Angel M.
Let ABC be a triangle and A'B'C' the cevian triangle of I.
Denote:
Oab, Oac = the circumcenters of AB'I, AC'I, resp.
Obc, Oba = the circumcenters of BC'I, BA'I, resp.
Oca, Ocb = the circumcenters of CA'I,CB'I, resp.
Oa, Ob, Oc = the circumcenters of IB'C', IC'A', IA'B', resp.
Denote:
Oab, Oac = the circumcenters of AB'I, AC'I, resp.
Obc, Oba = the circumcenters of BC'I, BA'I, resp.
Oca, Ocb = the circumcenters of CA'I,CB'I, resp.
Oa, Ob, Oc = the circumcenters of IB'C', IC'A', IA'B', resp.
Nab, Nac = the NPC centers of AB'I, AC'I, resp.
Nbc, Nba = the NPC centers of BC'I, BA'I, resp.
Nca, Ncb = the NPC centers of CA'I,CB'I, resp.
Na, Nb, Nc = the NPC centers of IB'C', IC'A', IA'B', resp.
2. The Euler lines of NaNabNac, NbNbcNba, NcNcaNcb are concurrent.
3. The parallels through A', B', C' to Euler lines of NaNabNac, NbNbcNba, NcNcaNcb, resp. are concurrent.
[Angel Montedeoca]:
**** 1 The Euler lines of OaOabOac, ObObcOba, OcOcaOcb are parallel with infinity point X(517)
**** 2. The Euler lines of NaNabNac, NbNbcNba, NcNcaNcb are concurrent in
(2 a^9 (b+c)+b c (b^2-c^2)^4-2 a^8 (b^2+c^2)+a (b-c)^4 (b+c)^3 (b^2+b c+c^2)-a^7 (7 b^3+b^2 c+b c^2+7 c^3)+2 a^2 (b^2-c^2)^2 (b^4-2 b^3 c-2 b c^3+c^4)+a^6 (6 b^4-2 b^3 c+8 b^2 c^2-2 b c^3+6 c^4)-a^3 (b-c)^2 (5 b^5+7 b^4 c+6 b^3 c^2+6 b^2 c^3+7 b c^4+5 c^5)+a^5 (9 b^5-4 b^4 c+b^3 c^2+b^2 c^3-4 b c^4+9 c^5)-a^4 (6 b^6-5 b^5 c+2 b^4 c^2+6 b^3 c^3+2 b^2 c^4-5 b c^5+6 c^6): ... : ... ).
The ETC-search numbers: (2.86418637501589, 3.22715415794361, 0.0845486610160832)
On lines {11,500},{30,1319},{496,5495},{511,6713},{549,4271},{952,5453}
**** 3. The parallels through A', B', C' to Euler lines of NaNabNac, NbNbcNba, NcNcaNcb, resp. are are concurrent in
(4 a^8 b c-2 a^9 (b+c)+b c (b^2-c^2)^4-12 a^6 b c (b^2+c^2)-6 a^2 b c (b^2-c^2)^2 (b^2+c^2)-a (b-c)^4 (b+c)^3 (b^2+3 b c+c^2)+a^7 (7 b^3+5 b^2 c+5 b c^2+7 c^3)+5 a^3 (b-c)^2 (b^5+3 b^4 c+2 b^3 c^2+2 b^2 c^3+3 b c^4+c^5)-a^5 (9 b^5+6 b^4 c-5 b^3 c^2-5 b^2 c^3+6 b c^4+9 c^5)-a^4 (-13 b^5 c+6 b^3 c^3-13 b c^5): ... : ... ).
The ETC-search numbers: (-0.0821492377402210, -1.14776402559181, 4.47318537858189)
On lines {11,8143}, {115,119}, {952,5492}, {1317,2771}
Angel M.
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