ADGEOM 3912

 

[Tran Quang Hung]:

 

Dear geometers,

Let ABC be a triangle with incenter I. 

P,Q are two isogonal conjugate points wrt ABC. 

 A*,B*,C* are isogonal conjugate of points A,B,C wrt triangle IPQ, respectively.

 Then A*,B*,C* and I are concyclic.

 Is this new circle ?

 

Best regards,

Tran Quang Hung.

 

[César Lozada]:

 

For P=u:v:w (trilinears), the center O(P) of the circle lies on the line PQ  and has coordinates:

O(P) = ((a+b-c)*a*c*v^2+(a-b+c)*a*b*w^2-(a+b-c)*(a-b+c)*a*v*w)*u^4+((a+b-c)*b*c*v^3+(a^3-(b+c)*a^2-(b^2+b*c-c^2)*a+(b-c)*(b^2-b*c+c^2))*w*v^2+(a^3-(b+c)*a^2+(b^2-b*c-c^2)*a-(b-c)*(b^2-b*c+c^2))*w^2*v+(a-b+c)*b*c*w^3)*u^3+(-(b-c)*(-a+b+c)*(a+b+c)*v^3*w+v*w^3*(b-c)*(-a+b+c)*(a+b+c))*u^2+(-(a-b+c)*b*c*v^4*w+(-a^3+(b+c)*a^2-(b^2-b*c-c^2)*a+(b-c)*(b^2-b*c+c^2))*w^2*v^3+(-a^3+(b+c)*a^2+(b^2+b*c-c^2)*a-(b-c)*(b^2-b*c+c^2))*w^3*v^2-(a+b-c)*b*c*v*w^4)*u+(a+b-c)*(a-b+c)*a*v^3*w^3-(a-b+c)*a*b*v^4*w^2-(a+b-c)*a*c*v^2*w^4 : :

ETC pairs (P,O(P)): (36,104), (80,104)

No simple relation among P, Q and O(P) was found.

Examples:

O(X(2)) = O(X(6)) = a^6-2*(b+c)*a^5-(b^2-6*b*c+c^2)*a^4-(b+c)*(b^2-3*b*c+c^2)*a^3-(2*b^4+2*c^4-b*c*(3*b^2-7*b*c+3*c^2))*a^2+(b+c)*(b^4+c^4+b*c*(3*b^2-7*b*c+3*c^2))*a-3*b*c*(b^2+c^2)*(b-c)^2 : : (trilinears)

= On lines: {2,6}, {111,9810}

= [ -0.473118610091199, 1.25326099907137, 2.991384687207831 ]

O(X(3)) = O(X(4)) = a^9-(b+c)*a^8-2*(b-c)^2*a^7+(b+c)*(3*b^2-5*b*c+3*c^2)*a^6-b*c*(6*b^2-11*b*c+6*c^2)*a^5-3*(b^3+c^3)*(b-c)^2*a^4+2*(b^4+c^4+2*b*c*(b^2+c^2))*(b-c)^2*a^3+(b^2-c^2)*(b-c)*(b^4+c^4-b*c*(b^2-4*b*c+c^2))*a^2-(b^2-c^2)^2*(b^4+c^4-b*c*(2*b-c)*(b-2*c))*a-(b^2-c^2)^3*(b-c)*b*c : :

= (3*r^2+4*R^2-s^2+4*R*r)*X(3)-R*(R-2*r)*X(4)

= Shinagawa coefficients: (4*r^2-2*F, 8*R*r-4*r^2-E+2*F)

= On lines: {2,3}, {74,759}, {104,900}, {477,12030}, {915,1295}, {953,2716}

= reflection of X(i) in X(j) for these (i,j): (4,867), (13589,3)

= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (140, 431, 1656), (468, 6983, 7501), (1592, 8359, 13154), (4243, 6617, 6898), (5192, 10299, 12108), (6097, 6906, 6930), (6918, 10257, 13745), (6946, 10021, 5020), (7428, 11308, 11484), (7442, 7456, 7429), (7485, 7866, 6143), (7557, 11328, 6855), (10989, 14014, 3538)

= [ 13.846312833877710, 12.94026598810948, -11.708587125496510 ]

 

César Lozada