HYACINTHOS 29279

[Antreas P. Hatzipolakis]:

Let ABC be a triangle and P a point.

Denote:

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

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

Which is the locus of P such that the reflections of L1, L2, L3 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 ={Neuberg cubic K001} ∪ {cubic q’, no ETC points on it}

q’ = ∑ [3*a^2*y*z*(c^2*(3*S^2-SC^2)*y+b^2*(3*S^2-SB^2)*z)] + 8*S^2*(2*S^2-3*SW*R^2)*x*y*z = 0

 

Some ETC-pair (P∈K001, Q(P)=point of concurrence): (1,1385), (3,5), (4,3), (13,13350), (14,13349), (15,6771), (16,6774), (30,12041), (74,30), (1138,110), (1157,12026), (1263,6150), (3065,10225)

 

If P ∈ K001 then Q(P) = midpoint(O, isogonal(P) )

 

Q( X(399) ) = MIDPOINT OF X(3) AND X(1138)

= 8*S^4-3*(9*R^2*(17*R^2+2*SA-8*SW)-4*SA^2+8*SW^2)*S^2+3*(27*(5*R^2-2*SW)*R^2+4*SW^2)*SB*SC : : (barys)

= 2*X(1511)+X(11749)

= lies on these lines: {3, 1138}, {30, 110}, {1511, 11749}, {5663, 18285}, {14677, 32417}, {16168, 31378}

= midpoint of X(3) and X(1138)

= [ 7.2868537808366040, 5.4617228841702300, -3.5036915675195950 ]

 

Q( X(484) ) = MIDPOINT OF X(3) AND X(3065)

= a*(2*a^9-3*(b+c)*a^8-2*(3*b^2-5*b*c+3*c^2)*a^7+2*(b+c)*(5*b^2-6*b*c+5*c^2)*a^6+6*(b^4+c^4-3*(b^2-b*c+c^2)*b*c)*a^5-(b+c)*(12*b^4+12*c^4-(24*b^2-29*b*c+24*c^2)*b*c)*a^4-2*(b^6+c^6-(3*b^2-b*c+3*c^2)*(b^2-b*c+c^2)*b*c)*a^3+3*(b^2-c^2)*(b-c)*(2*b^4+3*b^2*c^2+2*c^4)*a^2+2*(b^2-c^2)^2*(b-c)^2*b*c*a-(b^2-c^2)^4*(b+c)) : : (barys)

= (10*sin(A/2)-8*sin(3*A/2)+3*sin(5*A/2))*cos((B-C)/2)+2*(cos(A)-1)*cos(B-C)+2*sin(A)*cos(A/2)*cos(3*(B-C)/2)+cos(3*A) +6*cos(A)-5*cos(2*A) -7/2: : (trilinears)

= 3*X(21)-X(6265), 3*X(191)+X(6264), X(12119)-3*X(28460), X(12247)+3*X(15677)

= lies on these lines: {3, 3065}, {5, 26202}, {21, 104}, {80, 3579}, {191, 6264}, {214, 12104}, {517, 1749}, {952, 3647}, {1768, 7489}, {2800, 31649}, {2829, 22798}, {5441, 19914}, {5531, 32613}, {5533, 16142}, {5690, 22937}, {6713, 10021}, {6852, 16128}, {7508, 10176}, {7701, 32612}, {8068, 18977}, {10058, 16141}, {10074, 16140}, {10738, 16113}, {11684, 12737}, {12119, 28460}, {12247, 15677}, {12515, 21669}, {12611, 16617}

= midpoint of X(i) and X(j) for these {i,j}: {3, 3065}, {104, 3652}, {5441, 19914}, {10738, 16113}, {11684, 12737}, {12515, 21669}

= reflection of X(i) in X(j) for these (i,j): (214, 12104), (12611, 16617)

= [ 6.0108594153542110, 6.2946011980134950, -3.4913791545730610 ]

                          

César Lozada