HYACINTHOS 26481

 

[Antreas P. Hatzipolakis]:

Let ABC be a triangle and A'B'C' the cevian triangle of I.

Denote:
A", B", C" = the isogonal conjugates of I wrt triangles AB'C', BC'A', CA'B', resp.

A'B'C', A"B"C" are circumcyclologic.
(ie the circumcircles of A'B'C', A'B"C", B'C"A", C'A"B" are concurrent 

the circumcircles of A"B"C", A"B'C', B"C'A', C"A'B' are concurrent)

Cyclologic centers?

 

 

[César Lozada]:

 

Q(A’->A”)  = X(1)X(1263) ∩ X(11)X(137)

= (-a+b+c)*(a^6-(b^2+c^2)*a^4-( b^4+c^4+b*c*(2*b^2+b*c+2*c^2)) *a^2+(b^2-c^2)^2*(b+c)^2)*(a^ 3+(b+c)*a^2-(b^2-b*c+c^2)*a-( b^2-c^2)*(b-c))*(b-c)^2 : : (barycentrics

= (R-2*r)*X(11)+4*r*X(137)

= On the incentral circle and these lines: {1, 1263}, {11, 137}, {12, 6536}, {36, 12026}, {55, 11671}, {128, 3614}, {231, 13287}, {498, 13512}, {778, 4123}, {857, 6482}, {930, 5432}, {984, 11695}, {1020, 8010}, {1141, 7354}, {1245, 2274}, {1327, 9489}, {1475, 4286}, {1668, 3653}, {1715, 7188}, {1775, 2481}, {1823, 9412}, {1845, 7210}, {2560, 9550}, {2670, 9692}, {3898, 13370}, {4067, 6603}, {4788, 7286}, {5007, 7008}, {5054, 12269}, {5266, 7902}, {5326, 13372}, {5732, 11233}, {7157, 10652}, {7677, 7772}, {7951, 14072}, {8130, 10616}, {8248, 9112}, {8945, 12364}, {10592, 14073}, {10792, 13258}, {10819, 11036}, {11033, 13901}, {11356, 11849}

= {X(137), X(3327)}-Harmonic conjugate of X(11)

= [ -0.969473065463153, -0.30389215340899, 4.298500464481291 ]

 

Q(A”->A’) = X(35)X(500) ∩ X(36)X(6150)

= a^2*(a^2-b^2-b*c-c^2)*(a^6-2*( b^2+c^2)*a^4+(b^4+c^4+b*c*(b^ 2-b*c+c^2))*a^2-(b^2-c^2)^2*b* c)*(a^3+(b+c)*a^2-(b^2-b*c+c^ 2)*a-(b^2-c^2)*(b-c)) : : (barycentrics)

= R*X(1)+2*r*X(1157)

= On lines: {1, 1157}, {35, 500}, {36, 6150}, {241, 5194}, {544, 11872}, {949, 6957}, {1420, 3724}, {1432, 3308}, {1463, 13605}, {1562, 4670}, {1949, 2266}, {1956, 7339}, {2098, 9691}, {2143, 5330}, {2421, 11651}, {2486, 9237}, {2617, 13646}, {2790, 13450}, {2875, 9348}, {3051, 7288}, {3131, 11436}, {3342, 10197}, {3356, 7256}, {3607, 12922}, {3997, 12189}, {4217, 7322}, {5068, 13000}, {5103, 7301}, {5988, 10578}, {6482, 7200}, {6803, 8869}, {7876, 10507}, {8338, 10242}, {9289, 11758}, {9711, 11756}, {10780, 12854}

= [ 2.605243071958610, 3.26139379258185, 0.180356592292971 ]

 

César Lozada