[Kadir Altintas]
Let ABC be a triangle and P a point on the NPC,
Denote:
Ga, Gb, Gc = the G's (centroids) of PBC, PCA, PAB, resp.
The NPC of GaGbGc passes through G of ABC.
Which is its center for P on the NPC: X(11), X(115), X(125).... ?
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Let ABC be a triangle and P a point on the NPC,
Denote:
Ga, Gb, Gc = the G's (centroids) of PBC, PCA, PAB, resp.
The NPC of GaGbGc passes through G of ABC.
Which is its center for P on the NPC: X(11), X(115), X(125).... ?
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[Ercole Suppa]
Let W = W(P) the NPC center of GaGbGc.
We have:
W(X(11)) = MIDPOINT OF X(16173) AND X(26446) =
= 2 a^7-2 a^6 b-6 a^5 b^2+6 a^4 b^3+6 a^3 b^4-6 a^2 b^5-2 a b^6+2 b^7-2 a^6 c+8 a^5 b c-13 a^3 b^3 c+4 a^2 b^4 c+5 a b^5 c-2 b^6 c-6 a^5 c^2+8 a^3 b^2 c^2+2 a^2 b^3 c^2+2 a b^4 c^2-6 b^5 c^2+6 a^4 c^3-13 a^3 b c^3+2 a^2 b^2 c^3-10 a b^3 c^3+6 b^4 c^3+6 a^3 c^4+4 a^2 b c^4+2 a b^2 c^4+6 b^3 c^4-6 a^2 c^5+5 a b c^5-6 b^2 c^5-2 a c^6-2 b c^6+2 c^7 : : (barys)
= (4 a R^2-4 b R^2-4 a SB+4 b SB-c SB-4 a SC-b SC+4 c SC+4 a SW+b SW)S^2 -13 R S^3+3 R S SB SC+b SB SC^2-c SB SC^2-b SB SC SW : : (barys)
= 5*X[3]+X[10724], X[100]-7*X[3526], X[104]+5*X[1656], X[119]-4*X[3628], X[149]+11*X[3525], X[153]-13*X[5067], 5*X[631]+X[10738], 5*X[632]+X[1484], 2*X[1125]+X[12619], X[1385]+2*X[6702], X[1483]+2*X[3036], 5*X[1698]+X[12737], 7*X[3090]-X[10742], 4*X[3530]-X[24466], X[3579]+2*X[16174], 5*X[3616]+X[19914], 7*X[3624]-X[6265], 7*X[3851]-X[10728], 11*X[5056]+X[12248], 11*X[5070]+X[12773], 11*X[5550]+X[12247], X[5657]+3*X[32558], X[6174]-4*X[10124], X[6246]+2*X[13624], X[6684]+2*X[33709], 2*X[6701]+X[33856], X[6797]+2*X[31838], 5*X[8227]+X[12515], 2*X[9956]+X[11715], X[10265]+5*X[19862], 13*X[10303]-X[13199], X[10707]+5*X[15694], X[10711]-7*X[15703], X[11698]+2*X[20418], 2*X[15178]+X[15863], X[16173]+X[26446], 8*X[16239]-5*X[31235], 2*X[18254]+X[24475], X[19916]+5*X[30795]
W(X(11)) = MIDPOINT OF X(16173) AND X(26446) =
= 2 a^7-2 a^6 b-6 a^5 b^2+6 a^4 b^3+6 a^3 b^4-6 a^2 b^5-2 a b^6+2 b^7-2 a^6 c+8 a^5 b c-13 a^3 b^3 c+4 a^2 b^4 c+5 a b^5 c-2 b^6 c-6 a^5 c^2+8 a^3 b^2 c^2+2 a^2 b^3 c^2+2 a b^4 c^2-6 b^5 c^2+6 a^4 c^3-13 a^3 b c^3+2 a^2 b^2 c^3-10 a b^3 c^3+6 b^4 c^3+6 a^3 c^4+4 a^2 b c^4+2 a b^2 c^4+6 b^3 c^4-6 a^2 c^5+5 a b c^5-6 b^2 c^5-2 a c^6-2 b c^6+2 c^7 : : (barys)
= (4 a R^2-4 b R^2-4 a SB+4 b SB-c SB-4 a SC-b SC+4 c SC+4 a SW+b SW)S^2 -13 R S^3+3 R S SB SC+b SB SC^2-c SB SC^2-b SB SC SW : : (barys)
= 5*X[3]+X[10724], X[100]-7*X[3526], X[104]+5*X[1656], X[119]-4*X[3628], X[149]+11*X[3525], X[153]-13*X[5067], 5*X[631]+X[10738], 5*X[632]+X[1484], 2*X[1125]+X[12619], X[1385]+2*X[6702], X[1483]+2*X[3036], 5*X[1698]+X[12737], 7*X[3090]-X[10742], 4*X[3530]-X[24466], X[3579]+2*X[16174], 5*X[3616]+X[19914], 7*X[3624]-X[6265], 7*X[3851]-X[10728], 11*X[5056]+X[12248], 11*X[5070]+X[12773], 11*X[5550]+X[12247], X[5657]+3*X[32558], X[6174]-4*X[10124], X[6246]+2*X[13624], X[6684]+2*X[33709], 2*X[6701]+X[33856], X[6797]+2*X[31838], 5*X[8227]+X[12515], 2*X[9956]+X[11715], X[10265]+5*X[19862], 13*X[10303]-X[13199], X[10707]+5*X[15694], X[10711]-7*X[15703], X[11698]+2*X[20418], 2*X[15178]+X[15863], X[16173]+X[26446], 8*X[16239]-5*X[31235], 2*X[18254]+X[24475], X[19916]+5*X[30795]
= lies on these lines: {2,952}, {3,10724}, {5,2829}, {11,35}, {30,21154}, {100,3526}, {104,1656}, {119,3628}, {149,3525}, {153,5067}, {498,12735}, {499,1387}, {517,32557}, {528,11539}, {549,5840}, {631,10738}, {632,1484}, {1125,12619}, {1385,6702}, {1483,3036}, {1537,6952}, {1698,12737}, {2800,3833}, {2802,11231}, {3090,10742}, {3530,24466}, {3579,16174}, {3582,5844}, {3616,19914}, {3624,6265}, {3851,10728}, {5056,12248}, {5070,12773}, {5432,5533}, {5433,8068}, {5550,12247}, {5657,32558}, {6174,10124}, {6246,13624}, {6684,33709}, {6701,33856}, {6797,31838}, {6861,13226}, {6958,22791}, {7489,18861}, {7505,12138}, {7583,13977}, {7584,13913}, {8227,12515}, {8976,19081}, {9956,11715}, {10199,10283}, {10200,11729}, {10265,19862}, {10303,13199}, {10707,15694}, {10711,15703}, {11698,20418}, {13951,19082}, {15178,15863}, {16173,26446}, {16239,31235}, {18254,24475}, {19916,30795}
= midpoint of X(i) and X(j) for these {i,j}: {16173,26446}, {21154,23513}
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {11,140,33814}, {632,1484,3035}, {1125,12619,19907}, {6667,6713,5}
= (6-9-13) search numbers [3.3261232523588179353, 2.9831982670644505268, 0.0402395651587666144]
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W(X(115)) = COMPLEMENT OF X(15561) =
= 2 a^8-5 a^6 b^2+7 a^4 b^4-6 a^2 b^6+2 b^8-5 a^6 c^2+3 a^2 b^4 c^2-7 b^6 c^2+7 a^4 c^4+3 a^2 b^2 c^4+10 b^4 c^4-6 a^2 c^6-7 b^2 c^6+2 c^8 : : (barys)
= (3 SB SC-2 SB SW-2 SC SW-3 SW^2)S^2 + 11 S^4+SB SC SW^2 : : (barys)
= 5*X[3]+X[10723], X[4]-4*X[15092], X[98]+5*X[1656], X[99]-7*X[3526], X[114]-4*X[3628], X[115]+2*X[140], X[147]-13*X[5067], X[148]+11*X[3525], 2*X[547]+X[6055], X[549]+2*X[5461], 2*X[620]-5*X[632], 5*X[631]+X[6321], X[671]+5*X[15694], X[1511]+2*X[15359], X[2482]-4*X[10124], 7*X[3090]-X[6033], 7*X[3851]-X[10722], X[5054]+X[9166], 11*X[5056]+X[9862], 5*X[5071]+X[14830], X[5690]+2*X[11725], 2*X[5972]+X[15535], X[6054]-7*X[15703], 2*X[6721]+X[11623], 2*X[6723]+X[33511], X[9880]+2*X[12100], 2*X[9956]+X[11710], 13*X[10303]-X[13172], X[12117]-7*X[15701], 2*X[14693]+X[15980], 5*X[15059]+X[18332], 8*X[16239]-5*X[31274]
= lies on these lines: {2,2782}, {3,10723}, {4,15092}, {5,2794}, {30,5215}, {98,1656}, {99,3526}, {114,3628}, {115,140}, {147,5067}, {148,3525}, {542,15699}, {543,11539}, {547,6055}, {549,5461}, {620,632}, {631,6321}, {671,15694}, {1506,12829}, {1511,15359}, {2023,7746}, {2482,10124}, {2784,10172}, {3090,6033}, {3398,32967}, {3851,10722}, {4027,16922}, {5054,9166}, {5056,9862}, {5070,7943}, {5071,14830}, {5690,11725}, {5972,15535}, {6054,15703}, {6721,11623}, {6723,33511}, {7505,12131}, {7583,13967}, {7607,7934}, {7828,11272}, {7887,10104}, {7901,12176}, {7940,13108}, {9753,14881}, {9880,12100}, {9956,11710}, {10303,13172}, {12117,15701}, {14693,15980}, {15059,18332}, {16239,31274}
= midpoint of X(i) and X(j) for these {i,j}: {3,14639}, {5054,9166}, {14651,15561}
= reflection of X(22515) in X(14639)
= complement of X(15561)
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {2,14651,15561}, {5,6036,12042}, {5,12042,22505}, {115,140,33813}, {547,6055,22566}, {6036,6722,5}
= (6-9-13) search numbers [3.1866410325800904195, 2.9882098564828287196, 0.1011464123823790267]
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W(X(125)) = COMPLEMENT OF X(14643) =
= 2 a^10-5 a^8 b^2+a^6 b^4+7 a^4 b^6-7 a^2 b^8+2 b^10-5 a^8 c^2+12 a^6 b^2 c^2-10 a^4 b^4 c^2+9 a^2 b^6 c^2-6 b^8 c^2+a^6 c^4-10 a^4 b^2 c^4-4 a^2 b^4 c^4+4 b^6 c^4+7 a^4 c^6+9 a^2 b^2 c^6+4 b^4 c^6-7 a^2 c^8-6 b^2 c^8+2 c^10 :: (barys)
= (39 R^2+SB+SC-9 SW)S^2 + SB SC SW -9 R^2 SB SC :: (barys)
= 7*X[2]-X[5655], 2*X[3]+X[10113], X[4]-4*X[15088], 4*X[5]-X[1539], X[74]+5*X[1656], X[110]-7*X[3526], X[113]-4*X[3628], X[146]-13*X[5067], X[182]+2*X[6698], X[265]+5*X[631], X[381]+X[15055], 2*X[546]+X[16111], 2*X[548]+X[12295], X[550]+2*X[7687], 2*X[620]+X[15535], 10*X[632]-X[5609], 2*X[974]+X[5876], 2*X[1112]-5*X[15026], X[1353]+2*X[32257], 7*X[3090]-X[7728], 5*X[3091]+X[20127], X[3448]+11*X[3525], 7*X[3523]-X[12121], 2*X[3530]+X[11801], X[3581]+5*X[30745], 5*X[3763]+X[11579], 4*X[3850]-X[13202], 7*X[3851]-X[10721], 3*X[5054]-X[15035], 3*X[5055]+X[15041], 11*X[5056]+X[12244], 11*X[5070]+X[10620], 13*X[5079]+5*X[15021], 2*X[5092]+X[32274], 2*X[5447]+X[11800], 4*X[5498]-X[25487], X[5642]-4*X[10124], X[5690]+2*X[11735], 2*X[6684]+X[12261], 4*X[6689]-X[11702], X[7723]+2*X[13630], X[7731]-13*X[15028], X[9140]+5*X[15694], 2*X[9956]+X[11709], 4*X[10125]-X[20773], X[10263]-4*X[11746], 2*X[10272]+X[16003], 13*X[10303]-X[12383], 5*X[10574]+X[22584], X[10706]-3*X[15046], 4*X[11540]-X[11694], X[11557]-4*X[11695], 2*X[11793]+X[11806], X[11804]+2*X[32348], X[11805]-4*X[32396], X[12778]-7*X[31423], X[12825]-4*X[14128], 2*X[12900]+X[20417], X[12902]+5*X[15051], X[13201]+11*X[15024], X[13358]+2*X[32142], 2*X[13363]-X[16222], 4*X[13392]-X[24981], X[14651]+X[14850], X[15101]+2*X[25711], 2*X[15359]+X[33813], X[15647]+2*X[20299], X[16340]+2*X[22104], 2*X[20191]+X[33547], 2*X[20301]+X[33851], 2*X[32156]+X[32311]
= (6-9-13) search numbers [3.1866410325800904195, 2.9882098564828287196, 0.1011464123823790267]
-----------------------------------------------------------------------------
W(X(125)) = COMPLEMENT OF X(14643) =
= 2 a^10-5 a^8 b^2+a^6 b^4+7 a^4 b^6-7 a^2 b^8+2 b^10-5 a^8 c^2+12 a^6 b^2 c^2-10 a^4 b^4 c^2+9 a^2 b^6 c^2-6 b^8 c^2+a^6 c^4-10 a^4 b^2 c^4-4 a^2 b^4 c^4+4 b^6 c^4+7 a^4 c^6+9 a^2 b^2 c^6+4 b^4 c^6-7 a^2 c^8-6 b^2 c^8+2 c^10 :: (barys)
= (39 R^2+SB+SC-9 SW)S^2 + SB SC SW -9 R^2 SB SC :: (barys)
= 7*X[2]-X[5655], 2*X[3]+X[10113], X[4]-4*X[15088], 4*X[5]-X[1539], X[74]+5*X[1656], X[110]-7*X[3526], X[113]-4*X[3628], X[146]-13*X[5067], X[182]+2*X[6698], X[265]+5*X[631], X[381]+X[15055], 2*X[546]+X[16111], 2*X[548]+X[12295], X[550]+2*X[7687], 2*X[620]+X[15535], 10*X[632]-X[5609], 2*X[974]+X[5876], 2*X[1112]-5*X[15026], X[1353]+2*X[32257], 7*X[3090]-X[7728], 5*X[3091]+X[20127], X[3448]+11*X[3525], 7*X[3523]-X[12121], 2*X[3530]+X[11801], X[3581]+5*X[30745], 5*X[3763]+X[11579], 4*X[3850]-X[13202], 7*X[3851]-X[10721], 3*X[5054]-X[15035], 3*X[5055]+X[15041], 11*X[5056]+X[12244], 11*X[5070]+X[10620], 13*X[5079]+5*X[15021], 2*X[5092]+X[32274], 2*X[5447]+X[11800], 4*X[5498]-X[25487], X[5642]-4*X[10124], X[5690]+2*X[11735], 2*X[6684]+X[12261], 4*X[6689]-X[11702], X[7723]+2*X[13630], X[7731]-13*X[15028], X[9140]+5*X[15694], 2*X[9956]+X[11709], 4*X[10125]-X[20773], X[10263]-4*X[11746], 2*X[10272]+X[16003], 13*X[10303]-X[12383], 5*X[10574]+X[22584], X[10706]-3*X[15046], 4*X[11540]-X[11694], X[11557]-4*X[11695], 2*X[11793]+X[11806], X[11804]+2*X[32348], X[11805]-4*X[32396], X[12778]-7*X[31423], X[12825]-4*X[14128], 2*X[12900]+X[20417], X[12902]+5*X[15051], X[13201]+11*X[15024], X[13358]+2*X[32142], 2*X[13363]-X[16222], 4*X[13392]-X[24981], X[14651]+X[14850], X[15101]+2*X[25711], 2*X[15359]+X[33813], X[15647]+2*X[20299], X[16340]+2*X[22104], 2*X[20191]+X[33547], 2*X[20301]+X[33851], 2*X[32156]+X[32311]
= lies on these lines: {2,5655}, {3,10113}, {4,15088}, {5,1539}, {30,23515}, {74,1656}, {110,3526}, {113,3628}, {125,128}, {146,5067}, {182,6698}, {265,631}, {371,13979}, {372,13915}, {381,15055}, {541,15699}, {542,11539}, {546,16111}, {548,12295}, {549,17702}, {550,7687}, {620,15535}, {632,5609}, {974,5876}, {1112,15026}, {1353,32257}, {1493,26879}, {1986,6143}, {2931,7516}, {3043,13353}, {3090,7728}, {3091,20127}, {3448,3525}, {3523,12121}, {3530,11801}, {3548,6101}, {3581,30745}, {3763,11579}, {3850,13202}, {3851,10721}, {5054,15035}, {5055,15041}, {5056,12244}, {5070,10620}, {5079,15021}, {5092,32274}, {5447,11800}, {5498,25487}, {5642,10124}, {5690,11735}, {5892,10628}, {5944,10182}, {5965,14156}, {6102,6640}, {6684,12261}, {6689,11702}, {7505,12133}, {7583,13969}, {7723,13630}, {7731,15028}, {8976,19059}, {9140,15694}, {9540,19051}, {9826,15131}, {9956,11709}, {10125,20773}, {10263,11746}, {10272,16003}, {10303,12383}, {10574,22584}, {10706,15046}, {11540,11694}, {11557,11695}, {11793,11806}, {11804,32348}, {11805,32396}, {12292,14940}, {12778,31423}, {12825,14128}, {12900,20417}, {12902,15051}, {13201,15024}, {13358,32142}, {13363,16222} ,{13392,24981}, {13935,19052}, {13951,19060}, {14651,14850}, {15101,25711}, {15359,33813}, {15647,20299}, {15805,17847}, {16340,22104},{16657,23336}, {20191,33547}, {20301,33851}, {22804,32767}, {32156,32311}
= midpoint of X(i) and X(j) for these {i,j}: {2,15061}, {3,14644}, {381,15055}, {9140,32609}, {14651,14850}
= reflection of X(i) in X(j) for these {i,j}: {10113,14644}, {14644,20304}, {16222,13363}
= complement of X(14643)
= midpoint of X(i) and X(j) for these {i,j}: {2,15061}, {3,14644}, {381,15055}, {9140,32609}, {14651,14850}
= reflection of X(i) in X(j) for these {i,j}: {10113,14644}, {14644,20304}, {16222,13363}
= complement of X(14643)
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {3,15059,20304} ,{3,20304,10113}, {5,6699,12041}, {5,12041,1539}, {125,140,1511}, {632,10264,5972}, {632,20397,5609}, {3523,15081,12121}, {3530,11801,16163}, {5972,10264,5609}, {5972,20397,10264}, {6699,6723,5}, {12236,13416,6101}, {12902,15720,15051}
= (6-9-13) search numbers [3.6085021125044940502, 2.8855275481713006142, -0.0224709495209031914]
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Best regards,
Ercole Suppa
= (6-9-13) search numbers [3.6085021125044940502, 2.8855275481713006142, -0.0224709495209031914]
-----------------------------------------------------------------------------
Best regards,
Ercole Suppa
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