HYACINTHOS 28355

[Antreas P. Hatzipolakis]:

 

 
Let ABC be a triangle.

Denote:

Na, Nb, Nc = the NPC centers of IBC, ICA, IAB, resp.
A1, B1, C1 = points on IA, IB, IC such that:

AA1/AI = BB1/BI = CC1/CI = t

The perpendicular to AI at A1 intersects NbNc at A*
The perpendicular to BI at B1 intersects NcNa at B*
The perpendicular to CI at C1 intersects NaNb at C*

The points A*, B*, C* are collinear.

1. Which is the locus of the trilinear poles of the lines A*B*C* as t varies?
2. Which is the envelope of the lines A*B*C* as t varies?

[Angel Montesdsoca]:

***  2. The envelope of the lines A*B*C* as t varies is the parabola of barycentric equation:

b c ((a^10+a^9 (-b-c)+b^9 c-4 b^7 c^3+6 b^5 c^5-4 b^3 c^7+b c^9+a^8 (-4 b^2-b c-4 c^2)+a^7 (4 b^3+10 b^2 c+10 b c^2+4 c^3)+a^6 (6 b^4-3 b^3 c-7 b^2 c^2-3 b c^3+6 c^4)+a^5 (-6 b^5-18 b^4 c-17 b^3 c^2-17 b^2 c^3-18 b c^4-6 c^5)+a^4 (-4 b^6+10 b^5 c+29 b^4 c^2+51 b^3 c^3+29 b^2 c^4+10 b c^5-4 c^6)+a^3 (4 b^7+10 b^6 c-b^5 c^2-49 b^4 c^3-49 b^3 c^4-b^2 c^5+10 b c^6+4 c^7)+a^2 (b^8-7 b^7 c-18 b^6 c^2+11 b^5 c^3+42 b^4 c^4+11 b^3 c^5-18 b^2 c^6-7 b c^7+c^8)+a (-b^9-b^8 c+8 b^7 c^2+8 b^6 c^3-14 b^5 c^4-14 b^4 c^5+8 b^3 c^6+8 b^2 c^7-b c^8-c^9)) x^2-2 a^2 (-b^8+3 b^7 c-b^6 c^2-3 b^5 c^3+4 b^4 c^4-3 b^3 c^5-b^2 c^6+3 b c^7-c^8+a^6 (b^2-3 b c+c^2)+a^5 (2 b^2 c+2 b c^2)+a^4 (-3 b^4+11 b^3 c-22 b^2 c^2+11 b c^3-3 c^4)+a^3 (-2 b^4 c+4 b^3 c^2+4 b^2 c^3-2 b c^4)+a (-4 b^5 c^2+4 b^4 c^3+4 b^3 c^4-4 b^2 c^5)+a^2 (3 b^6-11 b^5 c+24 b^4 c^2-33 b^3 c^3+24 b^2 c^4-11 b c^5+3 c^6)) y z) +  ..... = 0

The focus of the parabola F: 

= MIDPOINT OF X(1) AND X(6788)

Barycentrics a^3 (b+c)-a^2 (-3 b^2+10 b c-3 c^2)+a (b^3+c^3)-(b^2-c^2)^2 ::

=  (r^2+13 r R-s^2) X(1)  - 3 r (2 r-R) X(2)   

= lies on these lines: {1,2}, {65,13756}, {244,21630}, {764,21201}, {946,3667}, {952,11717}, {1015,21090}, {1290,2718}, {2802,3756}, {3315,16173}, {4694,11813}, {11814,21087}, {12016,18240}, {18326,18493}.

= midpoint of X(1) and X(6788).
= reflection of X(i) in X(j) for these {i, j}:  {6789,1125}, {21087,11814}.

 (6 - 9 - 13) - search numbers  of F: (-0.287932454429568, 1.55842577853126, 2.69464622958408).
 
 The directrix of the parabola is the line X(1)X(88).
 
The point in the infinity of the parabola is X(2827) = isogonal conjugate of reflection (reflection of X(106) in line X(1)X(3)) in X(3).
 
 Angel Montesdeoca