1. Let ABC be a triangle.
Denote:
Na, Nb, Nc = the NPC centers of IBC, ICA, IAB, resp.
The circumcenter O' of NaNbNc is the N of ABC.
Let La, Lb, Lc be the Euler lines of O'NbNc, O'NcNa, O'NaNb, resp.
The parallels to La, Lb, Lc through A, B, C, resp. are concurrent
(at the reflection of I in Feuerbe4ch point ( = X(80))
Now we replace I with O and O' with I':
2. Let ABC be a triangle.
Denote:
Na, Nb, Nc = the NPC centers of OBC, OCA, OAB, resp.
The incenter I' of NaNbNc is the N of ABC.
Note: If ABC is not acute angled, then we take an excenter of NaNbNc instead of the incenter.
(if A > 90, then we take the Na-excenter I'a of NaNbNc)
Let La, Lb, Lc be the Euler lines of I'NbNc, I'NcNa, I'NaNb, resp.
The parallels to La, Lb, Lc through A, B, C, resp. are concurrent.
(Point? )
How to combine these two problems in order to get a locus problem?
This way:
Let ABC be a triangle and P a point.
Denote:
Na, Nb, Nc = the NPC centers of PBC, PCA, PAB, resp.
La, Lb, Lc = the Euler lines of NNbNc, NNcNa, NNaNb, resp.
Which is the locus of P such that the parallels to La, Lb, Lc through A, B, C, resp. are concurrent?
I, O lie on the locus.
[Ercole Suppa]:
Hi Antreas,
Problem 2. Suppose ABC is an acute-angled triangle. The parallels to La, Lb, Lc through A, B, C, resp. concur at the point:
Q1 = MIDPOINT OF X(13368) AND X(15101)
= (a^8-2 a^6 b^2+2 a^4 b^4-2 a^2 b^6+b^8-2 a^6 c^2+a^4 b^2 c^2+a^2 b^4 c^2-2 b^6 c^2-a^2 b^2 c^4+2 a^2 c^6+2 b^2 c^6-c^8) (a^8-2 a^6 b^2+2 a^2 b^6-b^8-2 a^6 c^2+a^4 b^2 c^2-a^2 b^4 c^2+2 b^6 c^2+2 a^4 c^4+a^2 b^2 c^4-2 a^2 c^6-2 b^2 c^6+c^8) : : (barys)
= (12 R^4+7 R^2 SB+7 R^2 SC-11 R^2 SW-2 SB SW-2 SC SW+2 SW^2)S^2 + 9 R^4 SB SC-9 R^2 SB SC SW+2 SB SC SW^2 : : (barys)
= 3*X[2]-2*X[11597], X[54]-2*X[125], X[110]-2*X[1209], 3*X[381]-2*X[11805], 2*X[1493]-5*X[15027], 4*X[6689]-5*X[15059], X[7728]-2*X[22804], 2*X[10113]-X[15800], 2*X[10610]-3*X[15061], X[10721]-2*X[32340], X[11562]-2*X[11802], X[11702]-2*X[20304], 2*X[11801]-X[20424], X[12307]+X[12902], X[13368]+X[15101], 4*X[13565]-3*X[14643], 3*X[15035]-4*X[32348], 2*X[15647]-X[32359], X[19150]-2*X[20301]
= lies on these lines: {2,11597}, {3,2888}, {4,7730}, {6,2914}, {30,11559}, {54,125}, {64,12244}, {67,32353}, {68,12319}, {70,32346}, {71,21092}, {74,10421}, {110,1209}, {146,18550}, {195,10224}, {265,1154}, {381,11805}, {399,3410}, {511,18125}, {539,5504}, {542,1176}, {895,5965}, {1173,3574}, {1352,20125}, {1493,15027}, {1899,3431}, {1987,15340}, {2777,16835}, {2781,15321}, {3024,12956}, {3028,12946}, {3527,7547}, {3531,18386}, {4846,11442}, {5663,6288}, {6145,6242}, {6391,31180}, {6689,15059}, {6776,19151}, {7687,14483}, {7691,17702}, {7728,22804}, {9920,13171}, {10113,15800}, {10274,14940}, {10610,15061}, {10721,32340}, {10821,11245}, {11270,11457}, {11271,15317}, {11562,11802}, {11702,20304}, {11801,20424}, {12307,12902}, {12903,18984}, {12904,13079}, {13198,32377}, {13202,13603}, {13368,15101}, {13418,21660}, {13423,18381}, {13452,14216}, {13472,18912}, {13565,14643}, {14542,32334}, {15035,32348}, {15041,15332}, {15089,16867}, {15100,18474}, {15110,18434}, {15647,32359}, {15738,22466}, {16665,20379}, {16868,17824}, {17835,18559}, {18377,21400}, {19150,20301}, {22336,32274}
= isogonal conjugate of X(2070)
= midpoint of X(13368) and X(15101)
= reflection of X(i) in X(j) for these {i,j}: {54,125}, {110,1209}, {195,11804}, {7728,22804}, {11562,11802}, {11702,20304}, {13417,11808}, {15800,10113}, {19150,20301}, {20424,11801}, {32359,15647}
= anticomplement of X(11597)
= antigonal image of X(54)= barycentric product X(3)*X(9381)
= barycentric quotient of X(i) and X(j) for these {i,j}: {6,2070}, {50,11597}, {184,9380}, {523,24978}, {2963,19552}, {3003,11557}, {9381,264}
= trilinear quotient of X(i) and X(j) for these {i,j}: {1,2070}, {1725,11557}, {6149,11597}
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {54,14076,6143}
= (6-9-13) search numbers: [13.1683024558331833615, -4.5948941219901385045, 0.7440669713622365163]
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Locus problem. Let P = (x:y:z) (barys) and denote Q = Q(P) the intersection point of La,Lb,Lc
Locus= {cicumcircle} U {K001 = Neuberg cubic}
*** Centers X(i) on locus for these i:
*** Pairs (P ∈ circumcircle, Q = Q(P)): {74,265}, {98,6321}, {99,6033}, {100,10742}, {101,10741}, {102,10747}, {103,10739}, {104,10738}, {105,15521}, {106,15522}, {107,22337}, {109,10740}, {110,7728}, {111,22338}, {112,12918}, {477,20957}, {930,31656}, {1113,10751}, {1114,10750}, {1291,14980}, {1292,10743}, {1293,10744}, {1294,10745}, {1295,10746}, {1296,10748}, {1297,10749}, {1300,13556}, {12092,22752}
*** Pairs (P ∈ K001, Q = Q(P)): {1,80}, {13,11602}, {14,11603}, {30,477}, {74,265}, {1263,25148}
*** Some points:
Q(X(3)) = Q1
Q(X(108)) = Q2 = REFLECTION OF X(3) IN X(25640)
= a^13-a^12 b-2 a^11 b^2+2 a^10 b^3-a^9 b^4+a^8 b^5+4 a^7 b^6-4 a^6 b^7-a^5 b^8+a^4 b^9-2 a^3 b^10+2 a^2 b^11+a b^12-b^13-a^12 c+7 a^11 b c-4 a^10 b^2 c-8 a^9 b^3 c+13 a^8 b^4 c-12 a^7 b^5 c-6 a^6 b^6 c+14 a^5 b^7 c-5 a^4 b^8 c+5 a^3 b^9 c+2 a^2 b^10 c-6 a b^11 c+b^12 c-2 a^11 c^2-4 a^10 b c^2+20 a^9 b^2 c^2-14 a^8 b^3 c^2-18 a^7 b^4 c^2+30 a^6 b^5 c^2-14 a^5 b^6 c^2+2 a^4 b^7 c^2+12 a^3 b^8 c^2-18 a^2 b^9 c^2+2 a b^10 c^2+4 b^11 c^2+2 a^10 c^3-8 a^9 b c^3-14 a^8 b^2 c^3+52 a^7 b^3 c^3-20 a^6 b^4 c^3-22 a^5 b^5 c^3+30 a^4 b^6 c^3-40 a^3 b^7 c^3+6 a^2 b^8 c^3+18 a b^9 c^3-4 b^10 c^3-a^9 c^4+13 a^8 b c^4-18 a^7 b^2 c^4-20 a^6 b^3 c^4+46 a^5 b^4 c^4-28 a^4 b^5 c^4-10 a^3 b^6 c^4+40 a^2 b^7 c^4-17 a b^8 c^4-5 b^9 c^4+a^8 c^5-12 a^7 b c^5+30 a^6 b^2 c^5-22 a^5 b^3 c^5-28 a^4 b^4 c^5+70 a^3 b^5 c^5-32 a^2 b^6 c^5-12 a b^7 c^5+5 b^8 c^5+4 a^7 c^6-6 a^6 b c^6-14 a^5 b^2 c^6+30 a^4 b^3 c^6-10 a^3 b^4 c^6-32 a^2 b^5 c^6+28 a b^6 c^6-4 a^6 c^7+14 a^5 b c^7+2 a^4 b^2 c^7-40 a^3 b^3 c^7+40 a^2 b^4 c^7-12 a b^5 c^7-a^5 c^8-5 a^4 b c^8+12 a^3 b^2 c^8+6 a^2 b^3 c^8-17 a b^4 c^8+5 b^5 c^8+a^4 c^9+5 a^3 b c^9-18 a^2 b^2 c^9+18 a b^3 c^9-5 b^4 c^9-2 a^3 c^10+2 a^2 b c^10+2 a b^2 c^10-4 b^3 c^10+2 a^2 c^11-6 a b c^11+4 b^2 c^11+a c^12+b c^12-c^13 : : (barys)
= 3*X[3]-4*X[6717], 2*X[5]-X[1295], 2*X[123]-3*X[381], 2*X[3627]-X[10731], 2*X[3845]-X[10715], X[10702]-2*X[22791], X[10763]-2*X[21850], X[10776]-2*X[22938], 2*X[11719]-X[18481], 4*X[11733]-5*X[18493]
= lies on these lines: {3,6717}, {4,280}, {5,1295}, {30,108}, {123,381}, {265,2778}, {382,2829}, {517,15499},{1359,1479}, {1478,3318}, {2791,6321}, {2798,6033}, {2804,10742}, {2812,10741}, {2817,10747}, {2823,10739}, {2834,15521}, {2840,15522}, {2845,22337}, {2849,10740}, {2850,7728}, {2851,22338}, {3627,10731}, {3845,10715}, {9521,10743}, {9525,10744}, {9528,10745}, {9531,10748}, {10702,22791}, {10763,21850}, {10776,22938}, {11719,18481}, {11733,18493}
= reflection of X(i) in X(j) for these {i,j}: {3,25640}, {1295,5}, {10702,22791}, {10715,3845}, {10731,3627}, {10746,4}, {10763,21850}, {10776,22938}, {18481,11719}
= (6-9-13) search numbers: [-12.5407430380077420933, -12.2971065549413068330, 17.9420811913319340306]
Q(X(399)) = Q3 = X(399)X(18319) ∩ X(1117)X(5663)
= (a^6-3 a^4 b^2+3 a^2 b^4-b^6-3 a^4 c^2-a^2 b^2 c^2+b^4 c^2+3 a^2 c^4+b^2 c^4-c^6) (a^20-4 a^18 b^2-3 a^16 b^4+48 a^14 b^6-126 a^12 b^8+168 a^10 b^10-126 a^8 b^12+48 a^6 b^14-3 a^4 b^16-4 a^2 b^18+b^20-4 a^18 c^2+19 a^16 b^2 c^2-47 a^14 b^4 c^2+85 a^12 b^6 c^2-101 a^10 b^8 c^2+49 a^8 b^10 c^2+31 a^6 b^12 c^2-53 a^4 b^14 c^2+25 a^2 b^16 c^2-4 b^18 c^2+9 a^16 c^4-17 a^14 b^2 c^4+45 a^12 b^4 c^4-111 a^10 b^6 c^4+160 a^8 b^8 c^4-195 a^6 b^10 c^4+165 a^4 b^12 c^4-53 a^2 b^14 c^4-3 b^16 c^4-24 a^14 c^6-23 a^12 b^2 c^6+75 a^10 b^4 c^6-19 a^8 b^6 c^6+107 a^6 b^8 c^6-195 a^4 b^10 c^6+31 a^2 b^12 c^6+48 b^14 c^6+54 a^12 c^8+25 a^10 b^2 c^8-140 a^8 b^4 c^8-19 a^6 b^6 c^8+160 a^4 b^8 c^8+49 a^2 b^10 c^8-126 b^12 c^8-72 a^10 c^10+25 a^8 b^2 c^10+75 a^6 b^4 c^10-111 a^4 b^6 c^10-101 a^2 b^8 c^10+168 b^10 c^10+54 a^8 c^12-23 a^6 b^2 c^12+45 a^4 b^4 c^12+85 a^2 b^6 c^12-126 b^8 c^12-24 a^6 c^14-17 a^4 b^2 c^14-47 a^2 b^4 c^14+48 b^6 c^14+9 a^4 c^16+19 a^2 b^2 c^16-3 b^4 c^16-4 a^2 c^18-4 b^2 c^18+c^20) (a^20-4 a^18 b^2+9 a^16 b^4-24 a^14 b^6+54 a^12 b^8-72 a^10 b^10+54 a^8 b^12-24 a^6 b^14+9 a^4 b^16-4 a^2 b^18+b^20-4 a^18 c^2+19 a^16 b^2 c^2-17 a^14 b^4 c^2-23 a^12 b^6 c^2+25 a^10 b^8 c^2+25 a^8 b^10 c^2-23 a^6 b^12 c^2-17 a^4 b^14 c^2+19 a^2 b^16 c^2-4 b^18 c^2-3 a^16 c^4-47 a^14 b^2 c^4+45 a^12 b^4 c^4+75 a^10 b^6 c^4-140 a^8 b^8 c^4+75 a^6 b^10 c^4+45 a^4 b^12 c^4-47 a^2 b^14 c^4-3 b^16 c^4+48 a^14 c^6+85 a^12 b^2 c^6-111 a^10 b^4 c^6-19 a^8 b^6 c^6-19 a^6 b^8 c^6-111 a^4 b^10 c^6+85 a^2 b^12 c^6+48 b^14 c^6-126 a^12 c^8-101 a^10 b^2 c^8+160 a^8 b^4 c^8+107 a^6 b^6 c^8+160 a^4 b^8 c^8-101 a^2 b^10 c^8-126 b^12 c^8+168 a^10 c^10+49 a^8 b^2 c^10-195 a^6 b^4 c^10-195 a^4 b^6 c^10+49 a^2 b^8 c^10+168 b^10 c^10-126 a^8 c^12+31 a^6 b^2 c^12+165 a^4 b^4 c^12+31 a^2 b^6 c^12-126 b^8 c^12+48 a^6 c^14-53 a^4 b^2 c^14-53 a^2 b^4 c^14+48 b^6 c^14-3 a^4 c^16+25 a^2 b^2 c^16-3 b^4 c^16-4 a^2 c^18-4 b^2 c^18+c^20) : : (barys)
= (3 R^2+2 SB+2 SC-2 SW) (2 S^4 + (-54 R^4-18 R^2 SC+6 SC^2+36 R^2 SW-6 SW^2)S^2-243 R^6 SC+324 R^4 SC^2-108 R^2 SC^2 SW+18 R^2 SC SW^2+6 SC^2 SW^2) (4 S^4 +(378 R^4+18 R^2 SB+6 SB SC+6 SC^2-144 R^2 SW-6 SB SW-6 SC SW+12 SW^2)S^2 + 243 R^6 SB+324 R^4 SB SC+324 R^4 SC^2-324 R^4 SB SW-324 R^4 SC SW-108 R^2 SB SC SW-108 R^2 SC^2 SW+90 R^2 SB SW^2+108 R^2 SC SW^2+6 SB SC SW^2+6 SC^2 SW^2-6 SB SW^3-6 SC SW^3) : : (barys)
= lies on these lines: {399,18319}, {1117,5663}
= antigonal image of X(3470)
= (6-9-13) search numbers: [-6.1466046337248925358, -6.3019948861517039928, 10.8404784648085861147]
Best regards,
Ercole Suppa
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