HYACINTHOS 26495

 

[Antreas P. Hatzipolakis]:

Let ABC be a triangle and P a point.

Denote:

A', B', C' = the reflections of P in A, B, C, resp.

 

Oa, Ob, Oc = the circumcenters of AB'C', BC'A', CA'B', resp.

O1, O2, O3 = the circumcenters of A'BC, B'CA, C'AB, resp.

Ga, G1 = the centroids of OaObOc, O1O2O3, resp.

 

For P = G:

Ga and G1 lie on the Euler line of ABC.

 

 

[César Lozada]:

For P=G:

 

Ga(P) = X(548)

 

G1(P) = Euler line intercept of X(3098)X(6144)

= 19*a^4-17*(b^2+c^2)*a^2-2*(b^ 2-c^2)^2 : :  (barycentrics)

= 2*X(2)-7*X(3) = 17*X(2)-7*X(4) = 19*X(2)-14*X(5) = 13*X(2)+7*X(20) = 3*X(2)+7*X(376) = 12*X(2)-7*X(381) = X(2)+4*X(548) = 9*X(2)-14*X(549) = 14*X(3098)+X(6144) = 14*X(3579)+X(3633)

= Shinagawa coefficients: (17, -21)

= On lines: {2, 3}, {3098, 6144}, {3579, 3633}, {3625, 3654}, {3635, 12702}, {3656, 12512}, {5023, 5346}, {5210, 5309}, {6446, 9541}, {6448, 9681}, {6496, 13903}, {6497, 13961}, {11592, 12290}, {11694, 12244}, {11850, 11999}

= midpoint of X(1656) and X(3534)

= reflection of X(i) in X(j) for these (i,j): (632, 12100), (3522, 8703), (3830, 3091), (3843, 2), (5071, 549)

= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (2, 3850, 5055), (3, 3830, 3524), (3, 5073, 3530), (376, 381, 3534), (381, 3526, 547), (547, 3830, 381), (631, 5076, 1656), (1656, 3843, 5072), (3091, 3526, 1656), (3135, 14065, 7462), (3524, 3830, 3526), (3839, 10299, 11812), (3839, 11812, 5070)

= [ 8.443560534879396, 7.55176628609364, -5.484509347640170 ]

 

The locus of P such that Ga lies on the Euler line of ABC is a quartic through ETC’s X(2)

The locus of P such that G1 lies on the Euler line of ABC is a quartic through ETC’s 2, 3413, 3414; these last ones in the infinity.

 

César Lozada