HYACINTHOS 29140

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle and P a point.

Denote:

Na, Nb, Nc = the NPC centers of PBC, PCA, PAB, resp.

La, Lb, Lc = the Euler lines of PBC, PCA. PAB. resp.

 

L1, L2, L3 = the parallels to La, Lb, Lc through P, resp.

Which is the locus of P such that the reflections of L1, L2, L3 in NbNc, NcNa, NaNb, resp. are concurrent?

Conjecture: Part of the locus is the Neuberg cubic.

For P = I, the point of concurrence is X(80)
Which are the points of concurrence for some other points P on the Neuberg cubic?

 

[César Lozada]: 

 

Locus = {Linf} ∪ {Nueberg cubic} ∪ {q4: degree-4 through X(4)} ∪  {q8: degree-8 through X(13), X(14), X(110)}

 

q4: ∑ [ y*z*((a^4-2*(b^2+c^2)*a^2+b^4+c^4)*x^2+a^4*y*z)] = 0

q8: ∑ [ y*z*((c^2*(2*a^6-2*(5*b^2+3*c^2)*a^4+(14*b^4+27*b^2*c^2+6*c^4)*a^2-(b^2+2*c^2)*(6*b^4+8*b^2*c^2+c^4))*y+b^2*(2*a^6-2*(3*b^2+5*c^2)*a^4+(6*b^4+27*b^2*c^2+14*c^4)*a^2-(2*b^2+c^2)*(b^4+8*b^2*c^2+6*c^4))*z)*x^5+(-c^2*(2*a^6+2*(3*b^2-5*c^2)*a^4-(18*b^4-3*b^2*c^2-14*c^4)*a^2+2*(b^2-c^2)*(5*b^4+7*b^2*c^2+3*c^4))*y^2+(3*a^8-8*(b^2+c^2)*a^6+(6*b^4+23*b^2*c^2+6*c^4)*a^4-8*(b^2+c^2)*b^2*c^2*a^2-(b^4+9*b^2*c^2+c^4)*(b^2-c^2)^2)*z*y-b^2*(2*a^6-2*(5*b^2-3*c^2)*a^4+(14*b^4+3*b^2*c^2-18*c^4)*a^2-2*(b^2-c^2)*(3*b^4+7*b^2*c^2+5*c^4))*z^2)*x^4+(13*a^8-16*(b^2+c^2)*a^6-(3*b^4-46*b^2*c^2+3*c^4)*a^4+2*(b^4-c^4)*(b^2-c^2)*a^2+4*(b^2-c^2)^4)*z^2*y^2*x^2-3*a^4*(a^4-(2*b^2-3*c^2)*a^2+(b^2-c^2)^2)*z^2*y^4+3*(3*a^4-(b^2+c^2)*a^2-2*(b^2-c^2)^2)*a^4*y^3*z^3-3*a^4*(a^4+(3*b^2-2*c^2)*a^2+(b^2-c^2)^2)*z^4*y^2)] = 0

 

ETC pairs (P,Q(P)) : (1, 80), (3, 10264), (30, 30), (74, 30), (1263, 24306)

 

Q( X(110) ) = MIDPOINT OF X(5) AND X(14611)

= 7*S^4-(3*R^2*(63*R^2+12*SA-34*SW)-8*SA^2+5*SB*SC+13*SW^2)*S^2-(9*R^2-SW)^2*SB*SC : : (barys)

= 3*X(110)+X(20957), X(477)+3*X(5655), X(12041)-3*X(31378), X(14480)+3*X(14643)

= lies on the cubic K913 and these lines: {5, 14611}, {30, 110}, {140, 14670}, {399, 16340}, {523, 10272}, {3258, 5609}, {3470, 3628}, {5663, 31379}, {12041, 31378}, {14480, 14643}, {14702, 18571}, {16168, 16534}

= midpoint of X(i) and X(j) for these {i,j}: {5, 14611}, {399, 16340}, {3258, 5609}, {10721, 21317}

= reflection of X(i) in X(j) for these (i,j): (140, 31945), (12079, 3628)

= [ 0.9637378444167425, -0.8447125127527643, 3.7806633702362580 ]

 

Q( X(15) ) = REFLECTION OF X(396) IN X(15609)

= (sqrt(3)*(SB+SC)+2*S)*(3*(15*R^2-2*SA-2*SW)*S^2-6*SB*SC*SW-S*sqrt(3)*(4*S^2-3*(SW+6*SA)*R^2+6*SA^2)) : : (barys)

= lies on these lines: {15, 5617}, {30, 8172}, {396, 15609}, {14446, 20579}

= reflection of X(396) in X(15609)

= [ 4.1819213561924280, 5.0774172357512640, -1.8045880764706930 ]

 

 

Q( X(16) ) = REFLECTION OF X(395) IN X(15610)

= (sqrt(3)*(SB+SC) -2*S)*(3*(15*R^2-2*SA-2*SW)*S^2-6*SB*SC*SW+S*sqrt(3)*(4*S^2-3*(SW+6*SA)*R^2+6*SA^2)) : : (barys)

= lies on these lines: {16, 5613}, {30, 8173}, {395, 15610}, {14447, 20578}

= reflection of X(395) in X(15610)

= [ -5.0828003303643830, -5..8352168075532220, 10.0263378088432500 ]

 

César Lozada