[Kadir Altintas]:
Let ABC be a triangle and A'B'C' the circumcevian triangle of X(2) = G.
The circle with diameter GA' intersects again the circumcircle at A"
The circle with diameter GA intersects again the circumcircle at A'"
Similarly B", B'" and C", C'"
1. AA", BB", CC" concur on the Euler line of ABC
2. A"A'", B"B'", C"C'" concur on the Euler line of ABC
3. The six centers of the circles lie on a circle cenetred on the midpoind of GO
We have also concurrences for X(4) = H instead of X(2) = G.
How about for X(5), X(20), X(140) ?
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[Ercole Suppa]
Let ABC be a triangle and A'B'C' the circumcevian triangle of P.
The circle with diameter PA' intersects again the circumcircle at A"
The circle with diameter PA intersects again the circumcircle at A'"
Similarly B", B'" and C", C'"
*** P = X(2) = G
1. AA", BB", CC" concur on the Euler on point X(26255)
2. A"A'", B"B'", C"C'" concur on the Euler line on point X(1995)
*** P = X(4) = H
1. AA", BB", CC" concur on the Euler on point X(24)
2. A"A'", B"B'", C"C'" concur on the Euler on point X(3542)
*** P = X(5) = N
1. AA", BB", CC" concur on the Euler on point X(13621)
2. A"A'", B"B'", C"C'" concur on point
Q1 = EULER LINE INTERCEPT OF X(54)X(20193)
= 4 a^10-9 a^8 b^2+2 a^6 b^4+8 a^4 b^6-6 a^2 b^8+b^10-9 a^8 c^2+4 a^6 b^2 c^2-9 a^4 b^4 c^2+17 a^2 b^6 c^2-3 b^8 c^2+2 a^6 c^4-9 a^4 b^2 c^4-22 a^2 b^4 c^4+2 b^6 c^4+8 a^4 c^6+17 a^2 b^2 c^6+2 b^4 c^6-6 a^2 c^8-3 b^2 c^8+c^10 : : (barys)
= (41 R^2-10 SW)S^2 + (R^2+6 SW)SB SC : : (barys)
Shinagawa coefficients: {e - 40 f, 25 e + 24 f}
= on lines X(i)X(j) for these {i,j}: {2,3},{54,20193},{5642,16982}
= reflection of X(i) in X(j) for these {i,j}: {5,21451}
= (6-8-13) search numbers [0.391102253613856809, -0.479449406904005356, 3.79208226194229505]
*** P = X(20)
1. AA", BB", CC" concur on the Euler on point X(11413)
2. A"A'", B"B'", C"C'" concur on point
Q2 = EULER LINE INTERCEPT OF X(69)X(11440)
= -5 a^10+9 a^8 b^2+2 a^6 b^4-10 a^4 b^6+3 a^2 b^8+b^10+9 a^8 c^2-32 a^6 b^2 c^2+18 a^4 b^4 c^2+8 a^2 b^6 c^2-3 b^8 c^2+2 a^6 c^4+18 a^4 b^2 c^4-22 a^2 b^4 c^4+2 b^6 c^4-10 a^4 c^6+8 a^2 b^2 c^6+2 b^4 c^6+3 a^2 c^8-3 b^2 c^8+c^10 : : (barys)
= (10 R^2-2 SW)S^2 + (-16 R^2+3 SW)SB SC : : (barys)
Shinagawa coefficients: {e - 4 f, -2 e + 6 f}
= on lines X(i)X(j) for these {i,j}: {2,3},{69,11440},{74,11411},{110,6225},{343,8567},{394,5894},{925,5897},{1092,20427},{1204,6515},{1294,13398},{5012,15740},{5504,16111},{5895,11064},{5925,20725},{9140,15077},{9833,16163},{11206,12279},{11441,12250},{12324,13445},{14457,18911},{15072,18925}
= reflection of X(i) in X(j) for these {i,j}: {4,3548},{3542,3}
= (6-8-13) search numbers [13.6146806694043087, 12.7092448751345968, -11.4417422790646381]
*** P = X(140)
1. AA", BB", CC" concur on the Euler on point X(22462)
2. A"A'", B"B'", C"C'" concur on point
Q3 = pending
= -10 a^10+21 a^8 b^2-2 a^6 b^4-20 a^4 b^6+12 a^2 b^8-b^10+21 a^8 c^2-18 a^6 b^2 c^2+47 a^4 b^4 c^2-53 a^2 b^6 c^2+3 b^8 c^2-2 a^6 c^4+47 a^4 b^2 c^4+82 a^2 b^4 c^4-2 b^6 c^4-20 a^4 c^6-53 a^2 b^2 c^6-2 b^4 c^6+12 a^2 c^8+3 b^2 c^8-c^10 : : (barys)
= (115 R^2-22 SW)S^2 + (-R^2+18 SW)SB SC : : (barys)
Shinagawa coefficients: {27 e - 88 f, 71 e + 72 f}
= on lines X(i)X(j) for these {i,j}: {2,3}
= (6-8-13) search numbers [0.893924693140423983, 0.0220465743309899432, 3.21282084130580577]
Best regards
Ercole Suppa
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