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
Particular case: P=O
Case 1: (Oa) is internally tangent to (O1)
Note: Barycentrics coordinates
Circle (O1):
· Center = -(a^3+(b+c)*a^2-(b+c)^2*a-(b+c)*(b^2+c^2))/(2*b*c) : b : c
· Squared-radius = (a+b-c)*(a-b+c)*b^2*c^2/((a+b+c)^3*(-a+b+c))
(O1) touchpoints:
· Ta = -a^3+2*(b^2-b*c+c^2)*a+(b^2-c^2)*(b-c) : b^2*(a+b-c) : c^2*(a-b+c)
· Ab = 1/c : 0 : 1/(a+b) = (O1) ∩ AC
· Ac = 1/b : 1/(a+c) : 0 = (O1) ∩ AB
Perspector ABC, TaTbTc = X(8)
Circle (Ta,Tb,Tc,H) through ETCs 3, 56 (antipodal points):
· Center = P1 = midpoint of X(3) and X(56)
· Squared-radius = R^3*(R-2*r)/(2*(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<=32609.
Case 2: (Oa) is externally tangent to (O1)
Note: Barycentrics coordinates
Circle (O1):
· Center = (a^3-(b+c)*a^2-(b+c)^2*a+(b+c)*(b^2+c^2))/(2*b*c) : b : c
· Squared-radius = (b-c+a)*(c+a-b)*b^2*c^2/((-a+b+c)^3*(a+b+c))
(O1) touchpoints:
· Ta = -a^3+2*(b^2-b*c+c^2)*a-(b^2-c^2)*(b-c) : b^2*(a+c-b) : c^2*(a+b-c)
· Ab = -1/c : 0 : 1/(a-b)
· Ac = -1/b : 1/(a-c) : 0
Perspector ABC, TaTbTc = X(55)
Circle (Ta,Tb,Tc,H) through ETCs 3, 55 (antipodal points):
· Center = P2 = midpoint of X(3) and X(55)
· Squared-radius = R^3*(R-2*r)/(2*(R+r))^2
Ab, Ac and their cyclic points are not co-conic.
---------------------------------------------------------------
Related centers:
P1 = MIDPOINT OF X(3) AND X(56)
= a^2*(a^5-(b+c)*a^4-2*(b-c)^2*a^3+(b+c)*(2*b^2-3*b*c+2*c^2)*a^2+(b^4+c^4-4*b*c*(b^2-b*c+c^2))*a-(b^3+c^3)*(b-c)^2) : : (barys)
= 3*X(3)-X(10310), 3*X(3)+X(10680), X(46)+3*X(3576), 3*X(56)+X(10310), 3*X(56)-X(10680), 5*X(631)-X(3436), 7*X(3523)+X(20076), 7*X(3526)-5*X(31246), 3*X(5054)-X(31141), 3*X(10165)-X(21616), X(11499)-3*X(16371), 3*X(21154)-X(32554)
= 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}, {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): (3, 22765, 40), (999, 11248, 24680), (3576, 7280, 3)
= [ 3.4847409586917350, 3.1720182911902510, -0.1636901621589041 ]
P2 = MIDPOINT OF X(3) AND X(55)
= a^2*(a^5-(b+c)*a^4-2*(b^2+c^2)*a^3+(b+c)*(2*b^2-b*c+2*c^2)*a^2+(b^4+c^4)*a-(b^2-c^2)*(b^3-c^3)) : : (barys)
= 3*X(3)-X(3428), 3*X(3)+X(10679), 3*X(55)+X(3428), 3*X(55)-X(10679), 5*X(631)-X(3434), 5*X(1656)-X(18499), X(3419)-3*X(26446), 7*X(3523)+X(20075), 7*X(3526)-5*X(31245), X(3870)+3*X(21165), 3*X(5054)-X(31140), 4*X(6690)-X(18407), 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}, {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): (3, 1482, 11012), (3, 10679, 3428), (3, 11248, 3579)
= [ 4.7089161551667640, 4.1828517521800070, -1.4285018796787480 ]
César Lozada
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