HYACINTHOS 28951

[Antreas P. Hatzipolakis]:

Let ABC be a triangle and L = the Euler line.

Denote:

Na, Nb, Nc = the NPC centers of NBC, NCA, NAB, resp

La, Lb, Lc = the parallels to L through Na, Nb, Nc, resp.

The reflections of La, Lb, Lc in NA, NB, NC, resp. are concurrent.

Point?  

 

[Peter Moses]:

Hi Antreas,

Point = MIDPOINT OF X(15345) AND X(20120)

2*a^16 - 11*a^14*b^2 + 21*a^12*b^4 - 9*a^10*b^6 - 25*a^8*b^8 + 43*a^6*b^10 - 29*a^4*b^12 + 9*a^2*b^14 - b^16 - 11*a^14*c^2 + 26*a^12*b^2*c^2 - 5*a^10*b^4*c^2 - 16*a^8*b^6*c^2 - 27*a^6*b^8*c^2 + 68*a^4*b^10*c^2 - 45*a^2*b^12*c^2 + 10*b^14*c^2 + 21*a^12*c^4 - 5*a^10*b^2*c^4 - 20*a^8*b^4*c^4 - 7*a^6*b^6*c^4 - 30*a^4*b^8*c^4 + 81*a^2*b^10*c^4 - 40*b^12*c^4 - 9*a^10*c^6 - 16*a^8*b^2*c^6 - 7*a^6*b^4*c^6 - 18*a^4*b^6*c^6 - 45*a^2*b^8*c^6 + 86*b^10*c^6 - 25*a^8*c^8 - 27*a^6*b^2*c^8 - 30*a^4*b^4*c^8 - 45*a^2*b^6*c^8 - 110*b^8*c^8 + 43*a^6*c^10 + 68*a^4*b^2*c^10 + 81*a^2*b^4*c^10 + 86*b^6*c^10 - 29*a^4*c^12 - 45*a^2*b^2*c^12 - 40*b^4*c^12 + 9*a^2*c^14 + 10*b^2*c^14 - c^16 : : 

 

= 3 X[5] - X[14143],3 X[5066] + X[23337]

 

= lies on these lines: {5, 252}, {30, 13856}, {113, 137}, {140, 15425}, {1154, 5501}, {5066, 20413}, {10272, 13362}, {11803, 14051}, {15345, 20120}, {18807, 19939}, {20030, 25150}

= midpoint of X(15345) and X(20120)

= reflection of X(140) and X(15425)

 

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5, 1157, 23280}, {546, 22051, 137}

Best regards,
Peter Moses.