ADGEOM 1893

Dear Friends:

 

Let (a’) be the parallel line to (BC) through A and Ab,Ac  the orthogonal projections of B,C on (a’). Build Bc,Cb, Ca, Cb cyclically.

 

Let P be a point and A’B’C’ the cevian triangle of ABC.

 

The circle {A’,Bc,Cb}  cuts (BC) again at A”. Build B”, C” cyclically.

 

Then lines (AA”), (BB”), (CC”) concur at Z(P)=X(1973)-Isoconjugate(P), i.e., for P:u:w:w (trilinears), they concur at:

 

Z(P) = (b^2+c^2-a^2)/(a^2*u) :  :
        = SA/((SB+SC)*u) : :

         =  cos(A)/(a^3*u) :  :

        =  X(1973)-Isoconjugate(P)

 

Some ETC-pairs (P,Z(P)) or (Z(P),P):

(1,304), (2,69), (3,76), (4,3926), (6,305), (7,345), (8,348), (21,1231), (48,561), 

 (57,3718), (63,75), (71,310), (72,274),  (77,312), (78,85),  (83,3933), (86,306),

 (92,326), (95,343), (97,311), (99,525)

 

Some non ETC Z(P):
Z(5)= SA/(a*(S^2+SB*SC)) : :

        = Isogonal conjugate of X(3199)

        = Isotomic conjugate of X(53)

        = ( 25.329144891733560, -11.33698411909894, -0.201028770285693 )

 

Z(9) = (b^2+c^2-a^2)/(a^2*(b+c-a)) : :

          = Isogonal conjugate of X(2212)

          = Isotomic conjugate of X(33)

          = ( 10.942446616969050, 4.50878797259950, -4.531164091570218 )

 

Z(10) = SA/(a*(b+c)) : :

           = Isogonal conjugate of X(2333)

          = Isotomic conjugate of X(1826)

          = ( 6.000643699474401, 2.68400656044699, -0.987021767390182 )

 

Regards

César Lozada