[Antreas P. Hatzipolakis]:
[Peter Moses]:
Hi Antreas,
Suppose L = {l1,l2,l3}, then
Radical trace = 2 a^2 (a^2+b^2-c^2) (a^2-b^2+c^2) l1^2-(2 a^6+3 a^4 b^2-4 a^2 b^4-b^6-3 a^4 c^2+4 a^2 b^2 c^2+3 b^4 c^2-3 b^2 c^4+c^6) l1 l2+(a^6+a^2 b^4-2 b^6-3 a^4 c^2+3 b^4 c^2+3 a^2 c^4-c^6) l2^2-(2 a^6-3 a^4 b^2+b^6+3 a^4 c^2+4 a^2 b^2 c^2-3 b^4 c^2-4 a^2 c^4+3 b^2 c^4-c^6) l1 l3+(3 a^4 b^2-6 a^2 b^4+3 b^6+3 a^4 c^2+4 a^2 b^2 c^2-3 b^4 c^2-6 a^2 c^4-3 b^2 c^4+3 c^6) l2 l3+(a^6-3 a^4 b^2+3 a^2 b^4-b^6+a^2 c^4+3 b^2 c^4-2 c^6) l3^2 : :
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Euler line -> X(3)
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OI line -> = X(3)X(513)∩X(4)X(8)
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Brocard axis -> = X(3)X(512)∩X(4)X(69)
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GK line -> X(15068).
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Orthic axis -> X(4)
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Anti-Orthic axis -> = X(1)X(3)∩X(4)X(513)
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Lemoine axis -> = X(3)X(6)∩X(4)X(512)
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De Longchamps line -> X(4).
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Gergonne line -> = X(3)X(142)∩X(4)X(514)
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Soddy line -> = X(3)X(514)∩X(4)X(9)
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Nagel line -> = X(3)X(3667)∩X(4)X(519)
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Fermat line -> = X(3)X(690)∩X(4)X(542)
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van Aubel line -> X(18338).
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IN line -> X(18342).
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Best regards,
Peter Moses.
Let ABC be a triange, A'B'C' the orthic triangle and L a line,
The parallels to L through A, B, C, intersect the circumcircle again at A", B", C", resp.
The cirumcircles of AA'A", BB'B", CC'C" are coaxial.
Radical trace ?
Special L = Euler line, Brocard axis, OI line.....
The parallels to L through A, B, C, intersect the circumcircle again at A", B", C", resp.
The cirumcircles of AA'A", BB'B", CC'C" are coaxial.
Radical trace ?
Special L = Euler line, Brocard axis, OI line.....
Hi Antreas,
Suppose L = {l1,l2,l3}, then
Radical trace = 2 a^2 (a^2+b^2-c^2) (a^2-b^2+c^2) l1^2-(2 a^6+3 a^4 b^2-4 a^2 b^4-b^6-3 a^4 c^2+4 a^2 b^2 c^2+3 b^4 c^2-3 b^2 c^4+c^6) l1 l2+(a^6+a^2 b^4-2 b^6-3 a^4 c^2+3 b^4 c^2+3 a^2 c^4-c^6) l2^2-(2 a^6-3 a^4 b^2+b^6+3 a^4 c^2+4 a^2 b^2 c^2-3 b^4 c^2-4 a^2 c^4+3 b^2 c^4-c^6) l1 l3+(3 a^4 b^2-6 a^2 b^4+3 b^6+3 a^4 c^2+4 a^2 b^2 c^2-3 b^4 c^2-6 a^2 c^4-3 b^2 c^4+3 c^6) l2 l3+(a^6-3 a^4 b^2+3 a^2 b^4-b^6+a^2 c^4+3 b^2 c^4-2 c^6) l3^2 : :
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Euler line -> X(3)
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= a (a^7 b^2-3 a^5 b^4+3 a^3 b^6-a b^8-2 a^6 b^2 c+a^5 b^3 c+5 a^4 b^4 c-2 a^3 b^5 c-4 a^2 b^6 c+a b^7 c+b^8 c+a^7 c^2-2 a^6 b c^2+2 a^5 b^2 c^2-a^4 b^3 c^2-5 a^3 b^4 c^2+4 a^2 b^5 c^2+2 a b^6 c^2-b^7 c^2+a^5 b c^3-a^4 b^2 c^3+4 a^3 b^3 c^3-a b^5 c^3-3 b^6 c^3-3 a^5 c^4+5 a^4 b c^4-5 a^3 b^2 c^4-2 a b^4 c^4+3 b^5 c^4-2 a^3 b c^5+4 a^2 b^2 c^5-a b^3 c^5+3 b^4 c^5+3 a^3 c^6-4 a^2 b c^6+2 a b^2 c^6-3 b^3 c^6+a b c^7-b^2 c^7-a c^8+b c^8) : :
= lies on these lines: {3,513},{4,8},{36,1935},{119,2818},{499,14115},{953,2975},{1329,31841},{2771,18341},{2779,21635},{2842,10265},{3025,5433},{3649,5462},{3874,13753},{3937,6713},{5687,15632},{6075,23154},{6830,30438},{8674,18342},{10575,18243},{10993,29349},{11491,14513},{15635,26492},{26470,29958}
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Brocard axis -> = X(3)X(512)∩X(4)X(69)
= a^2 (a^8 b^4-3 a^6 b^6+3 a^4 b^8-a^2 b^10-a^6 b^4 c^2+a^4 b^6 c^2-a^2 b^8 c^2+b^10 c^2+a^8 c^4-a^6 b^2 c^4+2 a^4 b^4 c^4-4 b^8 c^4-3 a^6 c^6+a^4 b^2 c^6+6 b^6 c^6+3 a^4 c^8-a^2 b^2 c^8-4 b^4 c^8-a^2 c^10+b^2 c^10) : :
= lies on these lines: {3,512},{4,69},{52,7843},{140,3111},{185,18347},{1078,2698},{1216,7873},{1975,15631},{2387,15980},{3091,6785},{4173,6071},{4230,9306},{5025,13137},{5462,15544},{6102,15536},{6784,20398},{7746,14113},{7936,7999}
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GK line -> X(15068).
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Orthic axis -> X(4)
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Anti-Orthic axis -> = X(1)X(3)∩X(4)X(513)
= a (a^7 b^2-3 a^5 b^4+3 a^3 b^6-a b^8-2 a^6 b^2 c+3 a^5 b^3 c+3 a^4 b^4 c-6 a^3 b^5 c+3 a b^7 c-b^8 c+a^7 c^2-2 a^6 b c^2+2 a^5 b^2 c^2-3 a^4 b^3 c^2+a^3 b^4 c^2+4 a^2 b^5 c^2-4 a b^6 c^2+b^7 c^2+3 a^5 b c^3-3 a^4 b^2 c^3+4 a^3 b^3 c^3-4 a^2 b^4 c^3-3 a b^5 c^3+3 b^6 c^3-3 a^5 c^4+3 a^4 b c^4+a^3 b^2 c^4-4 a^2 b^3 c^4+10 a b^4 c^4-3 b^5 c^4-6 a^3 b c^5+4 a^2 b^2 c^5-3 a b^3 c^5-3 b^4 c^5+3 a^3 c^6-4 a b^2 c^6+3 b^3 c^6+3 a b c^7+b^2 c^7-a c^8-b c^8) : :
= lies on these lines: {1,3},{4,513},{11,2818},{59,3562},{1046,2957},{1512,8679},{1519,2390},{1537,2841},{2771,18342},{2779,10265},{2817,12736},{2829,3937},{2842,21635},{3025,7354},{3109,18180},{3259,7681},{5552,15632},{5884,13868},{6073,12607},{6945,30438},{8674,18341},{11793,21677},{12114,15635},{15608,31841},{18242,23154}
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Lemoine axis -> = X(3)X(6)∩X(4)X(512)
= a^2 (a^8 b^4-3 a^6 b^6+3 a^4 b^8-a^2 b^10+a^6 b^4 c^2-3 a^4 b^6 c^2+3 a^2 b^8 c^2-b^10 c^2+a^8 c^4+a^6 b^2 c^4+2 a^4 b^4 c^4-2 a^2 b^6 c^4+2 b^8 c^4-3 a^6 c^6-3 a^4 b^2 c^6-2 a^2 b^4 c^6-2 b^6 c^6+3 a^4 c^8+3 a^2 b^2 c^8+2 b^4 c^8-a^2 c^10-b^2 c^10) : :
= lies on these lines: {3,6},{4,512},{51,1316},{184,21525},{185,18338},{381,18321},{1513,2387},{2698,12110},{2715,10312},{3060,4226},{3091,6787},{3150,13567},{3849,12508},{5562,7780},{5889,6179},{5890,7422},{6072,7764},{6784,11623},{6786,20399},{7752,12833},{7760,14510},{7763,15631},{7878,15043},{8704,13239},{12162,18348},{15026,15536},{15030,15098}
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De Longchamps line -> X(4).
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Gergonne line -> = X(3)X(142)∩X(4)X(514)
= 2 a^8-2 a^7 b+a^6 b^2-3 a^5 b^3-2 a^4 b^4+4 a^3 b^5+a^2 b^6+a b^7-2 b^8-2 a^7 c+3 a^5 b^2 c+3 a^4 b^3 c-6 a^2 b^5 c-a b^6 c+3 b^7 c+a^6 c^2+3 a^5 b c^2-2 a^4 b^2 c^2-4 a^3 b^3 c^2+3 a^2 b^4 c^2-3 a b^5 c^2+2 b^6 c^2-3 a^5 c^3+3 a^4 b c^3-4 a^3 b^2 c^3+4 a^2 b^3 c^3+3 a b^4 c^3-3 b^5 c^3-2 a^4 c^4+3 a^2 b^2 c^4+3 a b^3 c^4+4 a^3 c^5-6 a^2 b c^5-3 a b^2 c^5-3 b^3 c^5+a^2 c^6-a b c^6+2 b^2 c^6+a c^7+3 b c^7-2 c^8 : :
= lies on these linesL {3,142},{4,514},{664,10725},{3843,18329},{10767,18341}
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Soddy line -> = X(3)X(514)∩X(4)X(9)
= 2 a^8-2 a^7 b-a^6 b^2-a^5 b^3+4 a^3 b^5-a^2 b^6-a b^7-2 a^7 c+5 a^5 b^2 c+a^4 b^3 c-4 a^3 b^4 c-2 a^2 b^5 c+a b^6 c+b^7 c-a^6 c^2+5 a^5 b c^2-6 a^4 b^2 c^2+a^2 b^4 c^2+3 a b^5 c^2-2 b^6 c^2-a^5 c^3+a^4 b c^3+4 a^2 b^3 c^3-3 a b^4 c^3-b^5 c^3-4 a^3 b c^4+a^2 b^2 c^4-3 a b^3 c^4+4 b^4 c^4+4 a^3 c^5-2 a^2 b c^5+3 a b^2 c^5-b^3 c^5-a^2 c^6+a b c^6-2 b^2 c^6-a c^7+b c^7 : :
= lies on these lines: {3,514},{4,9},{20,18328},{103,3732},{191,2958},{220,3234},{348,14116},{1565,6712},{2328,4241},{2724,24047},{5532,6284},{10164,24980},{15634,17170}
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Nagel line -> = X(3)X(3667)∩X(4)X(519)
= 2 a^7-4 a^6 b+3 a^5 b^2+12 a^4 b^3-8 a^3 b^4-8 a^2 b^5+3 a b^6-4 a^6 c-6 a^4 b^2 c-9 a^3 b^3 c+13 a^2 b^4 c+9 a b^5 c-3 b^6 c+3 a^5 c^2-6 a^4 b c^2+14 a^3 b^2 c^2-a^2 b^3 c^2-3 a b^4 c^2-3 b^5 c^2+12 a^4 c^3-9 a^3 b c^3-a^2 b^2 c^3-18 a b^3 c^3+6 b^4 c^3-8 a^3 c^4+13 a^2 b c^4-3 a b^2 c^4+6 b^3 c^4-8 a^2 c^5+9 a b c^5-3 b^2 c^5+3 a c^6-3 b c^6 : :
= lies on these lines: {3,3667},{4,519},{3091,6788},{3146,6790},{7991,30196},{18341,21635}
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Fermat line -> = X(3)X(690)∩X(4)X(542)
= 2 a^14-6 a^12 b^2+12 a^10 b^4-21 a^8 b^6+21 a^6 b^8-9 a^4 b^10+a^2 b^12-6 a^12 c^2+4 a^10 b^2 c^2+5 a^8 b^4 c^2-6 a^6 b^6 c^2-2 a^4 b^8 c^2+8 a^2 b^10 c^2-3 b^12 c^2+12 a^10 c^4+5 a^8 b^2 c^4-12 a^6 b^4 c^4+9 a^4 b^6 c^4-25 a^2 b^8 c^4+9 b^10 c^4-21 a^8 c^6-6 a^6 b^2 c^6+9 a^4 b^4 c^6+32 a^2 b^6 c^6-6 b^8 c^6+21 a^6 c^8-2 a^4 b^2 c^8-25 a^2 b^4 c^8-6 b^6 c^8-9 a^4 c^10+8 a^2 b^2 c^10+9 b^4 c^10+a^2 c^12-3 b^2 c^12 : :
= lies on these lines: {3,690},{4,542},{5,5465},{98,15054},{99,15034},{110,23235},{125,18347},{541,10991},{543,30714},{631,11006},{2782,5609},{3090,18331},{3091,11005},{6055,20417},{11623,11656},{11638,25555},{13188,15039},{14639,15044},{14981,16534},{15027,15359},{15357,20397}
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van Aubel line -> X(18338).
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IN line -> X(18342).
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Best regards,
Peter Moses.
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