HYACINTHOS 26570

 [Antreas P. Hatzipolakis]:

Let ABC be a triangle, P a point and A'B'C' the pedal triangle of P.

Denote:

La, Lb, Lc = the Euler lines of AB'C', BC'A', CA'B', resp.

L1, L2, L3 = the reflections of La, Lb, Lc in PA', PB', PC', resp.

Li, Lii, Liii = the reflections of L1, L2, L3 in BC, CA, AB, resp.

1. P = gN = X54:

L1, L2, L3 are concurrent. Point?

 

2. P = N:

Li, Lii, Liii are concurrent. Point?

PS: We can ask for the loci. Are they complicated ?

[Angel Montesdeoca]:

1.  The locus of P such that L1, L2, L3 are concurrent is the   quartic of barycentric equation:

-a^2 b^10 c^6 x^3 y+b^12 c^6 x^3 y-a^8 b^2 c^8 x^3 y-a^6 b^4 c^8 x^3 y-a^4 b^6 c^8 x^3 y-a^2 b^8 c^8 x^3 y-5 b^10 c^8 x^3 y+4 a^6 b^2 c^10 x^3 y+7 a^4 b^4 c^10 x^3 y+9 a^2 b^6 c^10 x^3 y+10 b^8 c^10 x^3 y-6 a^4 b^2 c^12 x^3 y-11 a^2 b^4 c^12 x^3 y-10 b^6 c^12 x^3 y+4 a^2 b^2 c^14 x^3 y+5 b^4 c^14 x^3 y-b^2 c^16 x^3 y-a^8 b^4 c^6 x^2 y^2+a^4 b^8 c^6 x^2 y^2+a^6 b^4 c^8 x^2 y^2-a^4 b^6 c^8 x^2 y^2-a^12 c^6 x y^3+a^10 b^2 c^6 x y^3+5 a^10 c^8 x y^3+a^8 b^2 c^8 x y^3+a^6 b^4 c^8 x y^3+a^4 b^6 c^8 x y^3+a^2 b^8 c^8 x y^3-10 a^8 c^10 x y^3-9 a^6 b^2 c^10 x y^3-7 a^4 b^4 c^10 x y^3-4 a^2 b^6 c^10 x y^3+10 a^6 c^12 x y^3+11 a^4 b^2 c^12 x y^3+6 a^2 b^4 c^12 x y^3-5 a^4 c^14 x y^3-4 a^2 b^2 c^14 x y^3+a^2 c^16 x y^3+a^8 b^8 c^2 x^3 z-4 a^6 b^10 c^2 x^3 z+6 a^4 b^12 c^2 x^3 z-4 a^2 b^14 c^2 x^3 z+b^16 c^2 x^3 z+a^6 b^8 c^4 x^3 z-7 a^4 b^10 c^4 x^3 z+11 a^2 b^12 c^4 x^3 z-5 b^14 c^4 x^3 z+a^4 b^8 c^6 x^3 z-9 a^2 b^10 c^6 x^3 z+10 b^12 c^6 x^3 z+a^2 b^8 c^8 x^3 z-10 b^10 c^8 x^3 z+a^2 b^6 c^10 x^3 z+5 b^8 c^10 x^3 z-b^6 c^12 x^3 z+a^12 b^4 c^2 x^2 y z-4 a^10 b^6 c^2 x^2 y z+6 a^8 b^8 c^2 x^2 y z-4 a^6 b^10 c^2 x^2 y z+a^4 b^12 c^2 x^2 y z-a^12 b^2 c^4 x^2 y z+2 a^8 b^6 c^4 x^2 y z-a^4 b^10 c^4 x^2 y z+4 a^10 b^2 c^6 x^2 y z-2 a^8 b^4 c^6 x^2 y z-6 a^8 b^2 c^8 x^2 y z+4 a^6 b^2 c^10 x^2 y z+a^4 b^4 c^10 x^2 y z-a^4 b^2 c^12 x^2 y z-a^12 b^4 c^2 x y^2 z+4 a^10 b^6 c^2 x y^2 z-6 a^8 b^8 c^2 x y^2 z+4 a^6 b^10 c^2 x y^2 z-a^4 b^12 c^2 x y^2 z+a^10 b^4 c^4 x y^2 z-2 a^6 b^8 c^4 x y^2 z+a^2 b^12 c^4 x y^2 z+2 a^4 b^8 c^6 x y^2 z-4 a^2 b^10 c^6 x y^2 z+6 a^2 b^8 c^8 x y^2 z-a^4 b^4 c^10 x y^2 z-4 a^2 b^6 c^10 x y^2 z+a^2 b^4 c^12 x y^2 z-a^16 c^2 y^3 z+4 a^14 b^2 c^2 y^3 z-6 a^12 b^4 c^2 y^3 z+4 a^10 b^6 c^2 y^3 z-a^8 b^8 c^2 y^3 z+5 a^14 c^4 y^3 z-11 a^12 b^2 c^4 y^3 z+7 a^10 b^4 c^4 y^3 z-a^8 b^6 c^4 y^3 z-10 a^12 c^6 y^3 z+9 a^10 b^2 c^6 y^3 z-a^8 b^4 c^6 y^3 z+10 a^10 c^8 y^3 z-a^8 b^2 c^8 y^3 z-5 a^8 c^10 y^3 z-a^6 b^2 c^10 y^3 z+a^6 c^12 y^3 z+a^8 b^6 c^4 x^2 z^2-a^6 b^8 c^4 x^2 z^2+a^4 b^8 c^6 x^2 z^2-a^4 b^6 c^8 x^2 z^2+a^12 b^2 c^4 x y z^2-a^10 b^4 c^4 x y z^2+a^4 b^10 c^4 x y z^2-a^2 b^12 c^4 x y z^2-4 a^10 b^2 c^6 x y z^2+4 a^2 b^10 c^6 x y z^2+6 a^8 b^2 c^8 x y z^2+2 a^6 b^4 c^8 x y z^2-2 a^4 b^6 c^8 x y z^2-6 a^2 b^8 c^8 x y z^2-4 a^6 b^2 c^10 x y z^2+4 a^2 b^6 c^10 x y z^2+a^4 b^2 c^12 x y z^2-a^2 b^4 c^12 x y z^2+a^8 b^6 c^4 y^2 z^2-a^6 b^8 c^4 y^2 z^2-a^8 b^4 c^6 y^2 z^2+a^6 b^4 c^8 y^2 z^2+a^12 b^6 x z^3-5 a^10 b^8 x z^3+10 a^8 b^10 x z^3-10 a^6 b^12 x z^3+5 a^4 b^14 x z^3-a^2 b^16 x z^3-a^10 b^6 c^2 x z^3-a^8 b^8 c^2 x z^3+9 a^6 b^10 c^2 x z^3-11 a^4 b^12 c^2 x z^3+4 a^2 b^14 c^2 x z^3-a^6 b^8 c^4 x z^3+7 a^4 b^10 c^4 x z^3-6 a^2 b^12 c^4 x z^3-a^4 b^8 c^6 x z^3+4 a^2 b^10 c^6 x z^3-a^2 b^8 c^8 x z^3+a^16 b^2 y z^3-5 a^14 b^4 y z^3+10 a^12 b^6 y z^3-10 a^10 b^8 y z^3+5 a^8 b^10 y z^3-a^6 b^12 y z^3-4 a^14 b^2 c^2 y z^3+11 a^12 b^4 c^2 y z^3-9 a^10 b^6 c^2 y z^3+a^8 b^8 c^2 y z^3+a^6 b^10 c^2 y z^3+6 a^12 b^2 c^4 y z^3-7 a^10 b^4 c^4 y z^3+a^8 b^6 c^4 y z^3-4 a^10 b^2 c^6 y z^3+a^8 b^4 c^6 y z^3+a^8 b^2 c^8 y z^3=0

This quartic passes through the centers X (i), for i in {1, 3, 54, 195, 1147, 2574,  2575}.

For P = X(1), X(3), X(54),  L1, L2, L3 are concurrent in X(1), X(12041), X(10619), resp.

2.   The locus of P such that Li, Lii, Liii are concurrent is the  NAPOLEON CUBIC (K005 of Catalogue Bernard Gibert).

For P = X(1),  X(3),  X(54),  Li, Lii, Liii are concurrent in X(5903), X(30), X(10619),   resp.

 For P = X(4),  Li, Lii, Liii are concurrent in  the midpoint of X(7722) and X(7731) :
 
 (a^2 (2 a^12 (b^2+c^2)-4 a^10 (2 b^4+b^2 c^2+2 c^4)-(b^2-c^2)^4 (2 b^6+5 b^4 c^2+5 b^2 c^4+2 c^6)+2 a^8 (5 b^6+2 b^4 c^2+2 b^2 c^4+5 c^6)-a^4 (b^2-c^2)^2 (10 b^6+b^4 c^2+b^2 c^4+10 c^6)+a^6 (-11 b^6 c^2+10 b^4 c^4-11 b^2 c^6)+a^2 (8 b^12-13 b^10 c^2+10 b^6 c^6-13 b^2 c^10+8 c^12)) : ... : ....),
 
  and (6 - 9 - 13) - search numbers (1.60617721305135, 2.88615223652228, 0.901246527522174).
 
 For P = X(5),  Li, Lii, Liii are concurrent in  the midpoint of X(i) and X(j), for these {i, j}:  {5,6243}, {52,10263}, {143,13421}, {3627,5889},
 
  (a^2 (2 a^6 (b^2+c^2)-6 a^4 (b^4+b^2 c^2+c^4)-(b^2-c^2)^2 (2 b^4-b^2 c^2+2 c^4)+a^2 (6 b^6-b^4 c^2-b^2 c^4+6 c^6)) : ... : ...).
 
  and (6 - 9 - 13) - search numbers (-3.85861231142804, -2.93879602428523, 7.45611356553325).
 
  Angel Montesdeoca