HYACINTHOS 28487

 

[Antreas P. Hatzipolakis]:

 

 

Let ABC be a triangle and P a point.

 

Denote:

 

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

 

D = the Poncelet point of ABCP

 

H* = the orthocenter of NaNbNc

 

La. Lb, Lc = the reflections of DH* in NbNc, NcNa, NaNb, resp.

(concurrent on the circumcircle of NaNbNc)

 

For P = I, N:

The parallels to La, Lb, Lc through A, B, C, resp. are concurrent.

 

For P = I, the point of concurrence is the H

 

For P = N, which is the point of concurrence?

 

 

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[Ercole Suppa]

 

 

*** the parallels to La, Lb, Lc through A, B, C, resp. are concurrent at a ponint Q=Q(P) for every P in the plane

 

*** some points Q(X(i))

 

Q(X(1)) = X(4)

 

 

Q(X(2)) = ANTICOMPLEMENT OF X(16939)

 

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

= 3*X[2]-2*X[16939]

= lies on these lines: {2,16939},{8586,15533}

= anticomplement of X(16939)  

= antigonal image of isogonal conjugate of X(13492)

= (6-8-13) search numbers [7.92077140766388260, 1.12612318718194277, -0.794699912755680936]

 

 

Q(X(3)) = X(5964)

 

 

Q(X(4)) = X(3)

 

 

Q(X(5)) = X(143)

 

 

Q(X(6)) = ANTIGONAL IMAGE OF X(13493)

 

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

 

= antigonal image of X(13493)

 

= (6-8-13) search numbers [-3.45665132212365686, -0.406074740315288365, 5.51717068156742037]

 

 

Q(X(7)) = X(6)X(4845) ⋂ X(9)X(1155)

 

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

 

= on the cubics K220 and K950 and these lines : {6,4845},{9,1155}

 

= (6-8-13) search numbers [-1.68907540313982472, -1.46896592551811934, 5.43721415486915006]

 

 

Q(X(8)) = a^2 (3 a+3 b-5 c) (a+b-3 c) (a-b-c) (a-3 b+c) (3 a-5 b+3 c) (a^3-a^2 b-a b^2+b^3-a^2 c+8 a b c-3 b^2 c-a c^2-3 b c^2+c^3) : : (barys)

 

= on this line: {3680,5048}

 

= (6-8-13) search numbers [3.52135924345336481, 3.65873711202758047, -0.517550092628575751]

 

 

Best regards

Ercole Suppa