HYACINTHOS 29299

[Antreas P. Hatzipolakis]:

 

 

Variation of Hyacinthos 29279 

Let ABC be a triangle and P a point.

Denote:

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

La, Lb, Lc = the lines AA', BB', CC', resp.

La1, Lb1, Lc1 = the parallels to La, Lb, Lc through N, resp. (suggested notation)

Which is the locus of P such that the reflections of La1, Lb1, Lc1 in BC, CA, AB, resp. are concurrent?
O, H, I lie on the locus.

I think part of the locus is the Neuberg  cubic.


[César Lozada]:

 

 

Locus = The entire plane. (I hope I have not read it wrongly)

 

Q(P) = intersection point = Midpoint(O, P).

 

Q(X(35)) =  MIDPOINT OF X(3) AND X(35)

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

= 7*X(3523)+X(20066), 3*X(4995)+X(30264), 3*X(5054)-X(31159), X(5086)-3*X(26446), X(11491)+3*X(17549)

= lies on these lines: {1, 3}, {5, 20104}, {10, 7508}, {20, 10599}, {21, 9956}, {24, 1900}, {30, 31659}, {140, 25639}, {182, 9047}, {355, 4189}, {411, 28146}, {549, 3829}, {944, 17548}, {952, 5267}, {960, 22935}, {993, 32141}, {1012, 33697}, {2771, 33597}, {2779, 12041}, {3523, 20066}, {3530, 6713}, {3647, 31835}, {3828, 28463}, {4302, 6863}, {4330, 10738}, {4640, 5694}, {4995, 30264}, {5054, 31159}, {5086, 6875}, {5218, 10526}, {5248, 6924}, {5258, 12331}, {5303, 32900}, {5428, 6684}, {5432, 7491}, {5777, 22936}, {5886, 6942}, {6796, 6914}, {6842, 15338}, {6862, 18407}, {6892, 18517}, {6905, 9955}, {6906, 28160}, {6950, 18481}, {6954, 10525}, {11231, 25440}, {11491, 17549}, {11499, 16370}, {17100, 33598}, {19535, 22758}, {19925, 31649}, {22937, 31837}, {31660, 33595}

= midpoint of X(i) and X(j) for these {i,j}: {3, 35}, {6842, 15338}, {11012, 11849}

= reflection of X(i) in X(j) for these (i,j): (11011, 15178), (25639, 140), (33281, 1385)

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 26285, 3579), (1155, 24299, 5885), (3746, 22765, 33179)

= [ 5.0335262121576950, 4.4508907618945110, -1.7638873742461420 ]

 

César Lozada