HYACINTHOS 26542

[Le Viet An]:

Let ABC be a triangle, TaTbTc the pedal triangle of I and Ge the Gergonne point.
The line perpendicular to AI through Ge intersects AB, AC at Ab, Ac, resp.
Similarly Bc, Ba and Ca, Cb.
Let La, Lb, Lc be the Euler lines of GeAbBa, GeBcCb, GeCaAc, resp.

Then

1. La, Lb, Lc are concurrent.
2. The parallels to La, Lb, Lc through A, B, C, resp. are concurrent.
3. The parallels to La, Lb, Lc through Ta, Tb, Tc, resp.  are concurrent (on the incircle).

Points?

[César Lozada]:

 

 

1)      Z1 = X(1)X(651) ∩ X(7)X(528)

= a*(a^3-4*(b+c)*a^2+(5*b^2+3*b* c+5*c^2)*a-(b+c)*(2*b^2-b*c+2* c^2))*(a-b+c)*(a+b-c) : : (barys)

= 4*X(1)-X(1156) = X(100)+2*X(3243) = 4*X(142)-X(12531) = X(390)-4*X(12735) = 4*X(5083)-X(7672)

= on lines: {1, 651}, {7, 528}, {11, 3475}, {57, 100}, {59, 518}, {104, 2346}, {142, 12531}, {145, 10427}, {214, 1445}, {226, 10707}, {390, 6938}, {484, 10087}, {497, 12831}, {952, 1056}, {1145, 8732}, {1387, 8232}, {1388, 5220}, {1420, 6594}, {1617, 4430}, {2800, 7675}, {2802, 4321}, {3254, 10106}, {3315, 4551}, {3340, 5528}, {3600, 10609}, {3748, 10391}, {3889, 13279}, {4318, 4864}, {4326, 13253}, {4532, 5223}, {4861, 5784}, {5045, 12738}, {5119, 7676}, {5261, 12019}, {5289, 6172}, {5330, 5698}, {5425, 5542}, {5435, 6174}, {5572, 12740}, {5660, 11019}, {5722, 10711}, {5728, 6265}, {5732, 7673}, {5851, 8236}, {6049, 6068}, {6264, 9846}, {6326, 11025}, {7679, 10956}, {7982, 8544}, {8098, 8389}, {8104, 11234}, {8388, 12748}, {11219, 13405}, {12750, 13407}

= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (7, 1317, 12730), (1317, 3476, 10031)

= [ 0.739708258114655, 0.26983488603009, 3.112451903218324 ]

 

2)      Z2 = X(80)

3)      Z3 = X(1317) (on the incircle)

 

César Lozada