[Antreas P. Hatzipolakis]:
R1 = the radical axis of the circles (B*, B*B), (C*, C*C)
R2 = the radical axis of the circles (C*, C*C), (A*, A*A)
R3 = the radical axis of the circles (A*, A*A), (B*, B*B)
The point of concurrence Z depends on P and t but it lies always on the Euler line of ABC.
Addendum:
If P is such that OP/OH=q then Z(P,t) = Z(q,t) satisfies:
OZ(q,t) = OH*k, where k=((t-2)*(1-q)+2)/(t*(OH^2/R^ 2*q-1))
From this we deduce:
1) For q=R^2/OH^2 (i.e. P=X(186)), Z(P,t)=X(30).
2) For t=0, A*=B*=C*=P.
3) For q=0, ie P=O, Z(q,t) = X(20)
In general, trilinear coordinates of Z(q, t) are:
Z(q,t) = (q*t*(3+2*cos(2*A))-2*q-t)* cos(B-C)-q*t*(cos(3*A)+6*cos( A))+cos(A)*(4*q+3*t) : :
Let’s denote Z(q,t) as EHL(P,t).
Some EHL(P, t) (all trilinears) for t in {-1,-1/2,0,1/2,1}:
EHL(X(2),-1) = X(3) = O
EHL(X(2),-1/2) = X(378)
EHL(X(2),1/2)= (2*cos(2*A)-4)*cos(B-C)+11* cos(A)-cos(3*A) : :
= 2*X(3)-3*X(22) = 4*X(3)-3*X(378) = 5*X(3)-6*X(7502) = 3*X(3)-4*X(7555) = 5*X(22)-4*X(7502) = 9*X(22)-8*X(7555)
= Shinagawa coefficients: (2*E+2*F, -5*E-2*F)
= On lines: {2,3}, {159,5656}, {316,9723}, {511,11456}, {575,10984}, {576,7592}, {944,9911}, {1181,8718}, {1199,11482}, {1350,11459}, {1498,2781}, {1633,6361}, {3068,9695}, {3284,8743}, {3292,6759}, {4293,10833}, {4296,9645}, {8717,9730}, {10625,11441}
= reflection of X(378) in X(22)
= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (3,23,24), (3,1598,11284), (3,5198,3090), (3,7387,23), (3,7530,1995), (3,11284,631), (4,10323,7509), (4,11414,10323), (20,23,3), (20,7387,24), (26,1657,11413), (1995,7530,10594), (3146,7492,7527), (3529,7556,7464), (7464,7556,3), (7492,7527,3)
= [ -86.655074699301050, -87.29599634712987, 104.071004121905500 ]
EHL(X(2), 1) = (2*cos(2*A)-2)*cos(B-C)+7*cos( A)-cos(3*A) : :
= 3*X(3)-2*X(378) = 3*X(3)-4*X(7502) = 5*X(3)-8*X(7555) = 3*X(22)-X(378) = 3*X(22)-2*X(7502) = 5*X(22)-4*X(7555)
= Shinagawa coefficients: (3*E+4*F, -7*E-4*F)
= On lines: {2,3}, {35,9658}, {36,9673}, {115,8553}, {159,399}, {161,6000}, {195,11577}, {265,5621}, {394,10540}, {567,3796}, {999,4351}, {1154,11456}, {1181,6243}, {1351,8547}, {1482,9911}, {2917,5895}, {3070,9683}, {3098,5891}, {3295,4354}, {3579,8185}, {3581,10605}, {5446,10984}, {5889,8718}, {6101,11441}, {6449,8276}, {6450,8277}, {6759,10625}, {7592,10263}, {7737,9609}, {8148,8192}, {9655,10831}, {9659,10483}, {9668,10832}, {9914,9920}, {10564,11202}, {10620,11820}
= reflection of X(i) in X(j) for these (i,j): (3,22), (7391,5)
= Stammler circle-inverse-of-X(7574)
= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (3,1598,1656), (3,3843,7395), (3,5073,1593), (3,5899,25), (3,7387,7517), (3,7517,7506), (3,9909,2070), (4,6636,7514), (20,26,3), (22,378,7502), (23,376,6644), (25,5899,7517), (378,7502,3), (1657,2937,3), (3146,7512,7526), (3627,7525,7503), (5198,7393,3851), (7387,11414,3), (7503,7525,3), (7512,7526,3), (7556,11001,2071)
= [ -55.509262168135260, -56.23234725584278, 68.190410505861040 ]
EHL(X(4),-1) = (2*cos(2*A)+4)*cos(B-C)-7*cos( A)-cos(3*A) ::
= 3*X(3)-2*X(1658) = 3*X(3)-X(7387) = 15*X(3)-7*X(10244) = 17*X(3)-9*X(10245) = 3*X(5654)-X(5878)
= Shinagawa coefficients: (E-4*F, -3*E+4*F)
= On lines: {2,3}, {49,11456}, {52,1204}, {56,8144}, {64,155}, {74,5889}, {143,9786}, {156,1498}, {184,10575}, {394,5876}, {511,7689}, {542,9925}, {1069,10060}, {1092,10564}, {1147,6000}, {1151,11265}, {1152,11266}, {1154,10606}, {1236,1975}, {1288,1294}, {2883,9820}, {3157,10076}, {3357,9938}, {3796,10610}, {4299,9672}, {4302,9659}, {4550,11793}, {5204,9645}, {5446,11438}, {5584,8141}, {5621,11255}, {5654,5878}, {5946,10982}, {6102,10605}, {6759,12038}, {7747,9608}, {7756,9609}, {9730,11424}, {10263,12041}, {10539,11381}, {11267,11480}, {11268,11481}, {11412,11440}
= midpoint of X(64) and X(155)
= reflection of X(i) in X(j) for these (i,j): (3,11250), (26,3), (1498,156), (1658,10226), (2883,9820), (6759,12038), (7387,1658), (11477,11255)
= 1st Droz-Farny circle-inverse-of-X(403)
= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (3,4,6644), (3,382,24), (3,1593,5), (3,1597,6642), (3,5073,2070), (3,7387,1658), (3,7395,549), (3,7503,7516), (3,7517,186), (3,7526,7514), (4,2071,3), (4,3548,5), (24,382,7530), (186,3146,7517), (1597,6642,546), (1658,7387,26), (1658,10226,3), (2041,2042,11799), (7503,7516,7514), (7516,7526,7503), (7529,11403,3845)
= [ 15.681501822870570, 14.77061370221146, -13.822761230179200 ]
EHL(X(4),-1/2) = (2*cos(2*A)+6)*cos(B-C)-11* cos(A)-cos(3*A) ::
= 3*X(3)-2*X(26) = 5*X(3)-4*X(1658) = 11*X(3)-7*X(10244) = 13*X(3)-9*X(10245) = 3*X(3)-4*X(11250)
= Shinagawa coefficients: (E-2*F, -3*E+2*F)
= On lines: {2,3}, {36,9645}, {52,10605}, {56,9629}, {68,6247}, {154,12038}, {155,6000}, {511,3357}, {999,8144}, {1069,6285}, {1092,11381}, {1147,1498}, {1181,10575}, {1350,9973}, {1351,6102}, {1619,5878}, {1853,9927}, {1993,6241}, {2777,9914}, {2883,5654}, {2935,9937}, {3157,7355}, {3260,3964}, {3527,5946}, {4299,10832}, {4302,10831}, {4550,5447}, {5446,9786}, {5907,11472}, {6001,9928}, {6221,11265}, {6238,10060}, {6398,11266}, {6800,8718}, {7352,10076}, {7689,10606}, {8778,10317}, {9730,10982}, {9908,9938}, {10539,10564}
= reflection of X(i) in X(j) for these (i,j): (26,11250), (68,6247), (1498,1147), (7387,3), (9908,9938)
= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (3,4,6642), (3,382,25), (3,1597,5), (3,1598,6644), (3,3830,7506), (3,5073,7517), (3,7517,3515), (3,9714,186), (3,9909,1658), (4,3546,5), (4,7464,11413), (4,11413,3), (22,3520,3), (26,11250,3), (376,7503,3), (550,7526,3), (2071,3146,24), (3522,7527,7509), (3627,6644,1598), (3830,7506,5198), (9715,11410,3)
= [ 24.580640751544780, 23.64627647769151, -24.074745734130420 ]
EHL(X(4), 1/2) = X(7387)
EHL(X(4), 1) = X(26)
EHL(X(5), -1) = X(3520)
EHL(X(5),-1/2) = (2*cos(2*A)+5)*cos(B-C)-8*cos( A)-cos(3*A) ::
= 5*X(3)-3*X(2937) = 2*X(3)-3*X(3520) = 4*X(3)-3*X(7488)
= Shinagawa [E-4*F, -4*E+4*F]
= On lines: {2,3}, {52,74}, {54,10575}, {56,9539}, {64,1993}, {110,11381}, {185,1994}, {324,1105}, {511,11440}, {1204,3060}, {1498,9544}, {2935,3448}, {3357,5889}, {3580,6696}, {4550,7999}, {5584,9536}, {5866,7773}, {7355,9637}, {9306,11439}, {9545,11456}, {9786,11002}, {10539,11455}, {10574,11424}, {11003,11425}
= reflection of X(7488) in X(3520)
= 1st Droz-Farny circle-inverse-of-X(11563)
= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (3,3146,23), (3,3627,3518), (3,11403,1995), (382,11250,186), (1593,11413,2)
= [ 27.546846505014600, 26.60465730548796, -27.491873578052490 ]
EHL(X(5),1/2) = (2*cos(2*A)-3)*cos(B-C)+8*cos( A)-cos(3*A) ::
= 3*X(3)-5*X(2937) = 6*X(3)-5*X(3520) = 4*X(3)-5*X(7488)
= Shinagawa coefficients (3*E+4*F, -8*E-4*F)
= On lines: {2,3}, {52,8718}, {145,9911}, {161,6225}, {323,6759}, {3600,10833}, {7691,11381}, {8185,9778}
= reflection of X(3520) in X(2937)
= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (20,7387,23), (26,3529,2071), (382,7512,7527), (2937,3520,7488), (5198,7485,5068)
= [ -27.825109790500750, -28.62122637119619, 36.297718026812860 ]
EHL(X(5),1) = (2*cos(2*A)-1)*cos(B-C)+4*cos( A)-cos(3*A) ::
= 3*X(3)-5*X(2937) = 6*X(3)-5*X(3520) = 4*X(3)-5*X(7488)
= Shinagawa coefficients (2*E+4*F, -5*E-4*F)
= On lines: {2,3}, {110,10625}, {156,323}, {182,9781}, {185,8718}, {511,1614}, {515,9591}, {516,9626}, {575,1173}, {576,11423}, {1058,10833}, {1199,3060}, {1994,10263}, {2883,2917}, {2916,5480}, {2979,10539}, {3068,9683}, {3085,9658}, {3086,9673}, {3098,7999}, {3567,10984}, {3746,4354}, {4297,9625}, {4351,5563}, {5012,5446}, {5657,8185}, {6101,10540}, {6759,11412}, {7712,9545}, {7737,9700}, {8744,10316}, {9934,10628}
= reflection of X(7488) in X(2937)
= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (3,23,3518), (3,1995,3525), (3,3091,7550), (3,3627,7527), (3,3628,7496), (3,7530,3091), (3,7545,3628), (3,10594,3090), (4,22,7512), (20,26,186), (22,7387,4), (24,11414,376), (25,10323,631), (1598,7509,3545), (3091,7492,3), (3529,7556,3), (3547,7500,4), (3627,7555,3), (7485,7529,5067), (7492,7530,7550), (9909,11414,24)
= [ -13.982120716621910, -14.81475545202515, 20.350320125596520 ]
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