Let ABC be a triangle and P a point.
Denote:
(Oa), (Ob), (Oc) = the circles with diameters AP, BP, CP, resp.
(O1) = the circle tangent the sides AB, AC of the angle A and internally the circle (Oa) at Ta
(O2) = the circle tangent the sides BC, BA of the angle B and internally the circle (Ob) at Tb
(O3) = the circle tangent the sides CA, CB of the angle C and internally the circle (Oc) at Tc
Which is the locus of P such that:
1. The triangles ABC, TaTbTc are perspective?
2. The points P, Ta, Tb, Tc are concyclic?
H lies on the loci: Ercole Suppa, Romantics of Geometry 3128
How about for P = O ?
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[Ercole Suppa]:
Particular case: For P=O (barycentric coordinates)
Case 1: (Oa) is internally tangent to (O1)
*** Circle (O1):
Center = a^3-b^3-b^2 c-b c^2-c^3+a^2 (b+c)-a (b+c)^2 : -2 b^2 c :-2 b c^2
Squared-radius = b^2c^2(a+b-c)(a-b+c)/((-a+b+c)(a+b+c)^3)
(O1) touchpoints:
Ta = -a^3+(b-c)^2 (b+c)+2 a (b^2-b c+c^2) : b^2 (a+b-c) : c^2 (a-b+c)
Ab = a+b : 0 : c = (O1) ∩ AC
Ac = a+c : b : 0 = (O1) ∩ AB
*** Perspector ABC, TaTbTc = X(56)
*** Circle (Ta,Tb,Tc,O,X(56)) :
Center = a^2 (-a^5+2 a^3 (b-c)^2+a^4 (b+c)+a^2 (-2 b^3+b^2 c+b c^2-2 c^3)+(b-c)^2 (b^3+c^3)-a (b^4-4 b^3 c+4 b^2 c^2-4 b c^3+c^4)) : :
= 5*X[631]-X[3436], 7*X[3523]+X[20076], X[4299]+X[6928], 3*X[5054]-X[31141], X[11499]-3*X[16371]
= lies on these lines : {1,3},{4,10584},{5,2829},{10,32153},{12,21154},{24,1828},{30,7681},{84,31828},{104,355},{119,13747},{140,993},{182,8679},{214,5884},{388,6961},{474,9956},{496,5840},{499,6923},{515,6924},{528,32214},{529,549},{631,3436},{944,4188},{952,8256},{958,11231},{971,15297},{997,5694},{1006,5303},{1012,9955},{1125,6914},{1158,22775},{1478,6958},{1483,5854},{1656,18515},{1766,21773},{1837,10090},{2390,11202},{2975,6940},{3035,10942},{3086,6948},{3149,28160},{3523,20076},{3526,5251},{3560,3824},{3585,6971},{3616,6950},{3624,7489},{3653,17549},{3655,11491},{4190,10785},{4293,6891},{4299,6928},{4511,26877},{4640,31838},{4881,21740},{4973,31806},{5054,31141},{5229,6978},{5253,5886},{5267,10165},{5298,15908},{5322,16434},{5428,17768},{5433,6842},{5587,26321},{5690,8666},{5731,6942},{5818,17572},{5841,6922},{5881,12773},{5882,32141},{5887,17614},{6256,6959},{6265,18861},{6850,7288},{6875,11415},{6882,7354},{6885,18517},{6905,18481},{6909,12699},{6911,12114},{6918,18761},{6921,12115},{6929,10200},{6944,18516},{6955,10527},{6966,10532},{6970,12667},{6980,15446},{7491,15326},{7741,12764},{8227,13743},{8583,22936},{8703,12511},{9657,11929},{10058,11376},{10943,20418},{11112,26470},{11263,17009},{11499,16371},{12737,14923},{13731,27657},{14988,30144},{15325,15866},{18446,26201},{19525,19861},{19548,28096},{22753,22793},{22836,24475},{28459,30264}
= Midpoint of X(i) and X(j) for these {i,j}: {3,56},{4299,6928},{10310,10680}
= Reflection of X(i) in X(j) for these {i,j}: {5,6691},{1329,140}
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {1,3,26285}, {3,36,26286}, {3,55,26086}, {3,999,11248}, {3,1482,2077}, {3,3428,31663}, {3,10246,35}, {3,10269,1385}, {3,10680,10310}, {3,11249,3579}, {3,16202,5217}, {3,16203,55}, {3,22765,40}, {36,14800,1}, {55,16203,15178}, {56,8069,24928}, {56,10310,10680}, {104,404,355}, {474,22758,9956}, {997,24467,5694}, {999,1466,31794}, {999,11248,10222}, {1319,25414,1}, {1385,23961,3}, {1470,22766,942}, {2077,5563,1482}, {2975,6940,26446}, {3086,6948,10525}, {3560,25524,11230}, {3576,7280,3}, {4293,6891,10526}, {5253,6906,5886}, {5885,13624,26287}, {5885,26287,1}, {6911,12114,18480}, {8071,22768,24929}, {9940,13624,1385}, {13528,20323,23340},{15178,26086,55}, {31663,31797,3579}
=(6-9-13) search numbers: [3.48474095869173548801, 3.17201829119025209388, -0.16369016215890411019]
Squared-radius = R^2*(R-2*r)/4*(R-r)^2
*** Ab, Ac and their cyclic points lie on the conic K1 with center X(1001) and perspector X(10013). No ETC centers X(n) lie on K1 for n<=32000.
---------------------------------------
Case 1: (Oa) is externally tangent to (O1)
*** Circle (O1):
Center = a^3+b^3+b^2 c+b c^2+c^3-a^2 (b+c)-a (b+c)^2 : 2 b^2 c : 2 b c^2
Squared-radius = c^2*b^2(a+b-c)(a-b+c)/((-a+b+c)^3(a+b+c))
(O1) touchpoints:
Ta = -a^3-(b-c)^2 (b+c)+2 a (b^2-b c+c^2) : -b^2 (-a+b-c) : (a+b-c) c^2
Ab = -a+b : 0 : c = (O1) ∩ AC
Ac = -a+c : b : 0 = (O1) ∩ AB
*** Perspector ABC, TaTbTc = X(55)
*** Circle (Ta,Tb,Tc,O,X(55)) :
Center = a^2 (a^5-b^5+b^3 c^2+b^2 c^3-c^5-a^4 (b+c)-2 a^3 (b^2+c^2)+a^2 (2 b^3+b^2 c+b c^2+2 c^3)+a (b^4+c^4)) : :
= 2*X[140]-X[2886], 5*X[631]-X[3434], 7*X[3523]+X[20075], 7*X[3526]-5*X[31245], X[3870]+3*X[21165],X[4302]+X[6923], 3*X[5054]-X[31140], 2*X[10282]-X[10537], 3*X[16370]-X[22758]
= lies on these lines: {1,3},{4,10585},{5,5248},{8,6875},{10,32141},{11,21155},{12,7491},{21,355},{24,1824},{30,7680},{31,5396},{42,5398},{47,2594},{100,1006},{104,3655},{119,11113},{140,2886},{153,15677},{182,674},{255,5399},{405,9956},{411,12699},{495,5841},{497,6954},{498,6928},{500,601},{515,6914},{528,549},{529,32213},{631,3434},{902,1064},{912,4640},{943,5812},{944,4189},{952,993},{962,6876},{971,15296},{991,19624},{997,22935},{1001,6911},{1012,28160},{1030,2079},{1125,6924},{1283,14663},{1324,3185},{1376,6883},{1479,6863},{1483,5855},{1490,31828},{1621,5886},{1656,5259},{1871,14017},{2771,18446},{2875,18475},{3085,6868},{3145,9959},{3149,9955},{3523,20075},{3526,31245},{3560,11500},{3583,6980},{3584,15175},{3616,6942},{3651,16159},{3652,12528},{3653,13587},{3654,21161},{3679,12331},{3811,22937},{3870,21165},{3877,6265},{4192,29678},{4220,29665},{4294,6825},{4302,6923},{4420,26878},{4421,28466},{4428,22753},{4512,5720},{4995,28459},{4996,12737},{4999,10943},{5054,31140},{5070,25542},{5218,6827},{5251,5790},{5258,12645},{5267,5882},{5281,6987},{5284,6946},{5310,19544},{5428,5690},{5432,6882},{5534,31424},{5552,6936},{5587,7489},{5691,13743},{5694,12514},{5731,6950},{5762,8255},{5818,16865},{5840,6907},{5844,25439},{5887,20846},{6097,13754},{6253,6841},{6284,6842},{6824,18517},{6872,10786},{6902,27529},{6906,18481},{6910,12116},{6913,18491},{6917,10198},{6930,18516},{6962,10531},{6970,26105},{6985,11496},{7301,7609},{7330,22936},{7483,26470},{7488,20243},{7580,28146},{8053,14723},{9670,11928},{10282,10537},{11604,13199},{13323,22276},{15623,17524},{15733,31658},{15865,31789},{15888,30264},{16139,31660},{16370,22758},{19524,19861},{19649,29680},{20999,23206},{24309,29243},{24466,28458},{28208,28444}
= Midpoint of X(i) and X(j) for these {i,j}: {3,55}, {3428,10679}, {4302,6923}
= Reflection of X(i) in X(j) for these {i,j}: {5,6690}, {993,7508}, {2886,140}, {5173,13373},{10537,10282}, {18407,5}
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {1,3,26286}, {1,165,5535}, {1,1454,942} ,{3,35,26285}, {3,1482,11012}, {3,3295,11249}, {3,5217,26086}, {3,10246,36}, {3,10267,1385}, {3,10269,23961}, {3,10310,31663}, {3,10679,3428}, {3,11248,3579}, {3,11849,40}, {3,16202,56}, {3,16203,5204}, {21,11491,355}, {35,10902,3}, {35,14795,1}, {35,14799,5010}, {35,15931,2077}, {55,3428,10679}, {55,5172,1}, {55,8069,24929}, {56,16202,15178}, {100,1006,26446}, {405,11499,9956}, {1001,6911,11230}, {1376,6883,11231}, {1385,17502,18857}, {1385,23961,10269}, {1621,6905,5886}, {2077,10902,15931}, {2077,15931,3}, {3085,6868,10526}, {3295,11249,10222}, {3560,11500,18480}, {3576,5010,3}, {3746,11012,1482}, {3746,14804,1}, {4294,6825,10525}, {5248,6796,5}, {5267,5882,32153}, {6985,11496,22793}, {7489,18524,5587}, {8071,11510,24928}, {11508,26357,9957}, {13624,26086,3}, {26398,26422,1}
=(6-9-13) search numbers: [4.70891615516676473962, 4.18285175218000653815, -1.42850187967874761081]
Squared-radius = (R^3 (R-2r))/(4 (r + R)^2)
*** Ab, Ac and their cyclic points Ab, Ac and their cyclic points are not co-conic.
Best regards,
Ercole Suppa
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