HYACINTHOS 26544

 

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle and A'B'C', A"B"C" the cevian, circumcevian triangles of I, resp.

 

Denote:

A1, B1, C1 = the isogonal conjugates of A", B", C" wrt triangle A'B'C', resp.

 

A'B'C', A1B1C1 are perspective.
The perspector lies on the OI line.

 

 [César Lozada]:

 

 

Generalization:

 

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

 

Let A1, B1, C1 be the isogonal conjugates of A”, B”, C” w/r to A’B’C’,

 

A’B’C’ and A1B1C1 are perspective for any P.

 

 

For P=u:v:w (trilinears), the perspector Z(P) is

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

 

or, equivalently, for P=x:y:z (barycentrics)

Z(P) = (a^2*y^2*z^2+b^2*z^2*x^2+c^2* x^2*y^2+(-a^2+b^2+c^2)*x^2*y* z)*x : :

 

ETC pairs (P,Z(P)): (1,3746), (2,141), (4,4), (7,354), (100,1618)

 

Others:

 

Z( X(3) ) = X(3)X(49) ∩ X(30)X(1105)

= cos(A)*(4*cos(A)*cos(B-C)-cos( 2*(B-C))-cos(2*A)+cos(4*A)-3) : : (trilinears)

= (24*R^4-10*R^2*SW+S^2+SW^2)*X( 3)-2*(7*R^2-2*SW)*(4*R^2-SW)* X(49)

= On lines: {3, 49}, {4, 2055}, {30, 1105}, {417, 6760}, {418, 1614}, {426, 14059}, {542, 10600}, {577, 6759}, {933, 8439}, {1629, 13322}, {3284, 10110}, {6638, 10539}, {6641, 7592}, {9225, 9243}

= [ 66.783170974761210, 87.37871105976360, -87.675291317049440 ]

 

Z( X(6) ) =  X(6)X(25) ∩ X(54)X(695)

= a*(a^4-2*(b^2+c^2)*a^2-b^2*c^ 2) : : (trilinears)

= SW*(3*S^2+SW^2)*X(6)-2*S^2*(6* R^2-SW)*X(25)

= On lines: {2, 2056}, {6, 25}, {22, 13330}, {54, 695}, {111, 12834}, {182, 1613}, {323, 8041}, {511, 10329}, {524, 1799}, {575, 1196}, {1180, 11422}, {1207, 3203}, {1501, 11003}, {1627, 1691}, {1993, 3094}, {1994, 5111}, {2001, 13331}, {3117, 3398}, {3787, 5092}, {3796, 5017}, {3917, 5116}, {4074, 12215}, {5034, 9306}, {5104, 6636}, {5133, 11646}, {13410, 13595}

= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (2, 2056, 9225), (6, 184, 1915), (6, 9604, 1971), (1194, 13366, 6), (2056, 5038, 2), (3051, 5012, 1691)

= [ 1.416087480732205, 2.28767108413815, 1.403313355627716 ]

 

César Lozada