HYACINTHOS 24877

[Tran Quang Hung]:

 

Let ABC be a triangle.

 

Br1,Br2 are two Brocard points.

 

Let N1a,N1b,N1c are NPC center of triangles Br1BC,Br1CA,Br1AB.

 

Let N2a,N2b,N2c are NPC center of triangles Br2BC,Br2CA,Br2AB.

 

Then two triangles  N1aN1bN1c and  N2aN2bN2c are symmetric through a point.

 

Which is this point ?

 

 

[César Lozada]:

Triangles N1aN1bN1c and N2bN2cN2a are homothetic with homothetic center:

 

Z = Midpoint of {X(5), X(39)}

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

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

= (14*cos(2*A)+2*cos(4*A)-23)* cos(B-C)+(-8*cos(A)+4*cos(3*A) )*cos(2*(B-C))-cos(3*(B-C))+8* cos(3*A)-12*cos(A) :: (trilinears)

= midpoint of X(5) and X(39)

= reflection of X(i) in X(j) for these (i,j): (140,6683), (3934,3628)

= isogonal conjugate of X(592)

= On lines: {2,3095}, {3,83}, {5,39}, {6,10104}, {17,3106}, {18,3107}, {76,1656}, {140,143}, {147,6287}, {194,3090}, {538,547}, {549,5188}, {574,10358}, {576,7815}, {615,3103}, {631,9821}, {730,9956}, {732,7764}, {2021,7745}, {2080,7824}, {3091,7709}, {3097,8227}, {3329,3398}, {3525,6194}, {3628,3934}, {5055,7757}, {5097,7780}, {5790,7976}, {5976,7769}, {6036,7829}, {7393,9917}, {7736,9996}, {7808,9737}, {10346,10359}

= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (194,3090,7697), (262,7786,3)

= [ 1.058675694048923, 0.80310164713622, 2.596051482790571 ]

 

Regards,

César Lozada