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Antreas P. Hatzipolakis
Nab, Nac = the NPC centers of AB'G, AC'G, resp.
Nbc, Nba = the NPC centers of BC'G, BA'G, resp.
Nca, Ncb = the NPC centers of CA'G,CB'G, resp.
Na, Nb, Nc = the NPC centers of GB'C', GC'A', GA'B', resp.
G1, G2, G3 = the centroids of NaNabNac, NbNbcNba, NcNcaNcb
1. GaGbGc, G1G2G3 are perspective.
2.1 The Euler lines of OaOabOac, ObObcOba, OcOcaOcb are concurrent (parallels)
*** 1. GaGbGc, G1G2G3 are perspective, with perspector
(4 a^8+a^4 (4 b^4+7 b^2 c^2+4 c^4)-2 a^2 (3 b^6-5 b^4 c^2-5 b^2 c^4+3 c^6)-(b^2-c^2)^2 (2 b^4+7 b^2 c^2+2 c^4):...:...)
The ETC-search numbers: (-3.87333450089732, -5.55495472184388, 9.27409521282890)
On lines {2,1495},{4,3849},{30,7697},{98,381},{114,8592},{183,3830},{262,542},{3545,7694},{3839,9753},{3845,9993},{5066,7792},{6054,9830},{8370,9873}
*** 2.1 The Euler lines of OaOabOac, ObObcOba, OcOcaOcb are parallels with infinity point X(524)
**** 2.2. Let A*B*C* be the triangle bounded by the Euler lines of NaNabNac, NbNbcNba, NcNcaNcb.
ABC, A*B*C* are parallelogic.
The parallelogic center of triangle ABC with respect to a triangle A*B*C* is X(6094) = 11th Hatzipolakis-Montesdeoca Point.
The parallelogic center of triangle A*B*C* with respect to a triangle ABC is
(10 a^12-33 a^10 (b^2+c^2)-6 a^8 (11 b^4-34 b^2 c^2+11 c^4)+a^6 (221 b^6-108 b^4 c^2-108 b^2 c^4+221 c^6)-3 a^4 (41 b^8-112 b^6 c^2+288 b^4 c^4-112 b^2 c^6+41 c^8)-6 a^2 (5 b^10-28 b^8 c^2+8 b^6 c^4+8 b^4 c^6-28 b^2 c^8+5 c^10)+13 b^12-81 b^10 c^2+153 b^8 c^4-154 b^6 c^6+153 b^4 c^8-81 b^2 c^10+13 c^12 : ... : ...)
The ETC-search numbers: (2.95461451386555, 1.34981018911671, 1.34251226765794)
Angel M.
[APH]:
Let ABC be a triangle and A'B'C' the cevian triangle of G.
Denote:
Oab, Oac = the circumcenters of AB'G, AC'G, resp.
Obc, Oba = the circumcenters of BC'G, BA'G, resp.
Oca, Ocb = the circumcenters of CA'G,CB'G, resp.
Oa, Ob, Oc = the circumcenters of GB'C', GC'A', GA'B', resp.
Ga, Gb, Gc = the centroids of OaOabOac, ObObcOba, OcOcaOcbLet ABC be a triangle and A'B'C' the cevian triangle of G.
Denote:
Oab, Oac = the circumcenters of AB'G, AC'G, resp.
Obc, Oba = the circumcenters of BC'G, BA'G, resp.
Oca, Ocb = the circumcenters of CA'G,CB'G, resp.
Oa, Ob, Oc = the circumcenters of GB'C', GC'A', GA'B', resp.
Nab, Nac = the NPC centers of AB'G, AC'G, resp.
Nbc, Nba = the NPC centers of BC'G, BA'G, resp.
Nca, Ncb = the NPC centers of CA'G,CB'G, resp.
Na, Nb, Nc = the NPC centers of GB'C', GC'A', GA'B', resp.
G1, G2, G3 = the centroids of NaNabNac, NbNbcNba, NcNcaNcb
2.1 The Euler lines of OaOabOac, ObObcOba, OcOcaOcb are concurrent (parallels)
2.2. Let A*B*C* be the triangle bounded by the Euler lines of NaNabNac, NbNbcNba, NcNcaNcb.
ABC, A*B*C* are parallelogic. And also A'B'C', A*B*C*
[Angel Montesdeoca]:
*** 1. GaGbGc, G1G2G3 are perspective, with perspector
(4 a^8+a^4 (4 b^4+7 b^2 c^2+4 c^4)-2 a^2 (3 b^6-5 b^4 c^2-5 b^2 c^4+3 c^6)-(b^2-c^2)^2 (2 b^4+7 b^2 c^2+2 c^4):...:...)
The ETC-search numbers: (-3.87333450089732, -5.55495472184388, 9.27409521282890)
On lines {2,1495},{4,3849},{30,7697},{98,381},{114,8592},{183,3830},{262,542},{3545,7694},{3839,9753},{3845,9993},{5066,7792},{6054,9830},{8370,9873}
*** 2.1 The Euler lines of OaOabOac, ObObcOba, OcOcaOcb are parallels with infinity point X(524)
**** 2.2. Let A*B*C* be the triangle bounded by the Euler lines of NaNabNac, NbNbcNba, NcNcaNcb.
ABC, A*B*C* are parallelogic.
The parallelogic center of triangle ABC with respect to a triangle A*B*C* is X(6094) = 11th Hatzipolakis-Montesdeoca Point.
The parallelogic center of triangle A*B*C* with respect to a triangle ABC is
(10 a^12-33 a^10 (b^2+c^2)-6 a^8 (11 b^4-34 b^2 c^2+11 c^4)+a^6 (221 b^6-108 b^4 c^2-108 b^2 c^4+221 c^6)-3 a^4 (41 b^8-112 b^6 c^2+288 b^4 c^4-112 b^2 c^6+41 c^8)-6 a^2 (5 b^10-28 b^8 c^2+8 b^6 c^4+8 b^4 c^6-28 b^2 c^8+5 c^10)+13 b^12-81 b^10 c^2+153 b^8 c^4-154 b^6 c^6+153 b^4 c^8-81 b^2 c^10+13 c^12 : ... : ...)
The ETC-search numbers: (2.95461451386555, 1.34981018911671, 1.34251226765794)
Angel M.
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