HYACINTHOS 28972

[Vu Thanh Tung]:

 

Dear all

 

Let P be the de Longchamps point (X(20)) of a triangle ABC.

 

P_AP_BP_C is the pedal triangle of P w.r.t ABC.

 

X_A,X_B,X_C are the orthopoles of the lines P_BP_C, P_CP_A, P_AP_B w.r.t ABC.

 

 

then triangles X_AX_BX_C and ABC are perspective. What is the perspector ?

 

 

Apart from X(20) and the circumcenter (ABC), is there any other point having this property ?

 

Best regards,

 

Vu Thanh Tung

 

 

[César Lozada]:

 

 

Locus = {Linf} ∪  { circumcircle} ∪ {cubic q3 through ETC’s 3,20,1498}

q3 = ∑ [((SA+SB)*(-3*SC^3*SA+(2*SC^2-SC*SA+SB*SA)*S^2)*y-(SA+SC)*(-3*SB^3*SA+(2*SB^2-SB*SA+SC*SA)*S^2)*z)*x^2-SB*SC*b^2*c^2*x^3*(SB-SC)]  -S^2*x*y*z*(SB-SC)*(SC-SA) *(SA-SB) = 0 (barys)

 

Perspectors Q(P):

Q(X(3)) = X(4)

 

Q( X(20) ) = X(4)X(253) ∩ X(25)X(64)

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

= SB*SC*(SB+SC)*(S^2+4*(4*R^2-SW)*(SA+SC))*(S^2+4*(4*R^2-SW)*(SA+SB)) : : (barys)

= lies on these lines: {4, 253}, {25, 64}, {185, 3343}, {235, 459}, {1073, 1593}, {1301, 3515}, {3516, 14379}, {5907, 15394}, {6524, 6526}, {6622, 14572}, {8798, 11403}, {11589, 15750}

= {X(64), X(28785)}-harmonic conjugate of X(1204)

≈ [ -4.2890471409389890, -5.4429084151565190, 9.3883920651415010 ]

 

Q0=Q(X(1498)) = X(2)X(3349) ∩ X(4)X(3344)

= (S^2-2*(4*R^2-SB)*SB)*(3*S^2-16*R^2*(4*R^2+SB-2*SW)+4*SB^2-2*SC*SA-4*SW^2)*(S^2-2*(4*R^2-SC)*SC)*(3*S^2-16*R^2*(4*R^2+SC-2*SW)-4*SW^2-2*SA*SB+4*SC^2) : : (barys)

= lies on Kiepert hyperbola and these lines: {2, 3349}, {4, 3344}, {459, 3346}

= isogonal conjugate of Q0*

= {X(2), X(14365)}-harmonic conjugate of X(3349)

≈ [ -0.4688374111641116, -0.4851934802544232, 4.1929541580054930 ]

 

whose isogonal conjugate is:
Q0* = X(3)X(6) ∩ X(1498)X(3343)

= (SB+SC)*(S^4-2*SB*SC*(S^2+SA^2))*(S^6-2*SB*SC*(S^4+2*(S^2-SB*SC)*SA^2)) : : (barys)

= lies on these lines: {3, 6}, {1498, 3343}, {2060, 3350}, {6621, 13567}

= isogonal conjugate of Q0

≈ [ 3.5738595900023330, 3.4533833331027160, -0.3996130210108850 ]

 

César Lozada