HYACINTHOS 26166

[Antreas P. Hatzipolakis]:

Let ABC be a triangle and A'B'C' the pedal triangle of H.

Denote:

N1 = the NPC center of AB'C'

Na = the reflection of N1 in HA'. Similarly Nb, Nc.

1. ABC, NaNbNc are orthologic.

 

2. The centroid of  NaNbNc lies on the Euler line.

[Peter Moses]:

 

Hi Antreas,

 

1)

(ABC, NaNbNc) orthology: 

(a^8+a^6 b^2-4 a^4 b^4+a^2 b^6+b^8-3 a^6 c^2+4 a^4 b^2 c^2+4 a^2 b^4 c^2-3 b^6 c^2+3 a^4 c^4-4 a^2 b^2 c^4+3 b^4 c^4-a^2 c^6-b^2 c^6) (a^8-3 a^6 b^2+3 a^4 b^4-a^2 b^6+a^6 c^2+4 a^4 b^2 c^2-4 a^2 b^4 c^2-b^6 c^2-4 a^4 c^4+4 a^2 b^2 c^4+3 b^4 c^4+a^2 c^6-3 b^2 c^6+c^8)::

X(11270)-vertex conjugate of X(11816).

Searches: {34. 3195110499112839860690677134, 32. 7459574088450243863509985002,- 34. 8693880549443073117719051115}.

 

(NaNbNc, ABC) orthology: X(13474).

 

2).

2 a^10-3 a^8 b^2-2 a^6 b^4+4 a^4 b^6-b^10-3 a^8 c^2-6 a^4 b^4 c^2+6 a^2 b^6 c^2+3 b^8 c^2-2 a^6 c^4-6 a^4 b^2 c^4-12 a^2 b^4 c^4-2 b^6 c^4+4 a^4 c^6+6 a^2 b^2 c^6-2 b^4 c^6+3 b^2 c^8-c^10:: 

on lines {{2,3},{143,12134},{206, 5476},{539,11808},{542,9969},{ 1503,5946},{3564,9971},{5462, 13419},{6102,11745},{6146, 10095},{10110,12370},{12824, 13451}}.

Midpoint of X(i) and X(j) for these {i,j}: {{2, 7540}, {381, 7576}, {7553, 7667}}.

Reflection of X(i) in X(j) for these {i,j}: {{549, 10127}, {7667, 140}}.

(J^2 - 11) X[2] - (J^2 - 7) X[3].

2 X[546] + X[3575], X[1885] - 4 X[3861], 5 X[3843] + X[6240], X[5] + 2 X[6756], 2 X[140] + X[7553], 5 X[632] - 4 X[7734], X[550] - 4 X[9825], X[6146] - 4 X[10095], 2 X[10691] - 3 X[11539], X[6102] - 4 X[11745], 4 X[6756] - X[11819], 2 X[5] + X[11819], 2 X[143] + X[12134], 7 X[3851] - X[12225], 5 X[5] - 2 X[12362], 5 X[6756] + X[12362], 5 X[11819] + 4 X[12362], 4 X[10110] - X[12370], 4 X[3850] - X[12605], X[7667] - 8 X[13163], X[140] - 4 X[13163], X[7553] + 8 X[13163], 2 X[5462] + X[13419].

{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,7506,13371),(5,26,7568),(5, 6756,11819),(25,381,10201),( 25,11818,5),(26,7528,5),(381, 7426,547),(381,10201,5),(546, 10020,5576),(546,10096,13413), (2070,5133,140),(3518,5576, 10020),(3542,7564,5),(6997, 7514,5),(10201,11818,381).

inverse of X(5189) in the first Droz-Farney circle.

Searches: {-0. 692982667055058937788198586398 ,-1. 56067448417940973636997384308, 5. 04096958651861304573827557220} .

 

Best regards,

Peter Moses.