HYACINTHOS 25674

[Antreas P. Hatzipolakis]:

Let ABC be a triangle and P a point.

Denote:

A'B'C' = the cevian triangle of P

A"B"C" = the circumcevian triangle of P.

Ab, Ac = the orthogonal projections of A' on AC, AB, resp.

A2, A3 = the orthogonal projections of A" on A'Ab, A'Ac, resp.

Similarly B3, B1 and C1, C2

 

La, Lb, Lc = the Euler lines of A"A2A3, B"B3B1, C"C1C2, resp.

L1, L2, L3 = the parallels to La, Lb, Lc through A, B, C, resp.

 

Which is the locus of P such that:

1. La, Lb, Lc are concurrent?

2. L1, L2, L3 are concurrent?

 

[César Lozada]:

 

1.       Degree-12-circumcurve

2.       Napoleon-Feuerbach cubic

[APH]:

H lies on the 1.

Point of concurrence?

[César Lozada]:

1)      Z1(I) = I; Z1(O) = X(30)

 

Z1(H) = SB*SC*(SB+SC)*((18*R^2+6*SA-5* SW)*S^2+(36*R^2-11*SW)*SA^2) : : (barycentrics)

= 2*X(4)-3*X(1112) = X(4)-3*X(1986) = X(4)+3*X(7722) = 4*X(4)-3*X(12133) = 5*X(4)-3*X(12292) = 4*X(140)-3*X(12358) = 3*X(185)-X(10990) = 4*X(389)-3*X(12099) = X(1112)+2*X(7722) = 5*X(1656)-3*X(7723)

= midpoint of X(1986) and X(7722)

= reflection of X(i) in X(j) for these (i,j): (1112,1986), (7723,9826), (12133,1112)

= On lines: {4,94}, {24,5609}, {74,3516}, {110,3515}, {113,12359}, {125,12233}, {140,12358}, {185,1205}, {389,12099}, {399,3517}, {541,1885}, {542,3575}, {974,10628}, {1154,10295}, {1593,11482}, {1656,7723}, {2777,6146}, {2914,10610}, {3523,12219}, {3542,5655}, {5094,5890}, {7507,9140}, {10018,11561}, {10019,11557}, {10294,11591}

= [ 6.660646391513457, 7.74000010312901, -4.791941616341915 ]

 

2)      ETC-pairs (P,Z2(P)):

(1,1), (3,30), (4,74), (5,3), (17,15), (18,16), (54,4), (61,13), (62,14), (195,1263), (627,1337), (628,1338), (2120,3482), (2121,3481), (3336,3065), (3459,1157), (3460,3483), (3461,7165), (3462,3484), (3463,8439), (3467,484), (3468,3465), (3469,3466), (3470,1138), (3471,399), (3489,3479), (3490,3480), (6191,1277), (6192,1276), (7344,7060), (7345,7059), (8837,8174), (8839,8175), (8918,5668), (8919,5669), (8929,8446), (8930,8456)

 

The locus of Z2(P) seems to be the Neuberg cubic.

 

César Lozada