HYACINTHOS 27517

[Antreas P. Hatzipolakis]:

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

Denote:

A1, B1, C1 = the orthogonal projections of A, B, C on OA', OB', OC', resp.

A2, B2, C2 = the orthogonal projections of A', B', C' on OA, OB, OC, resp

Na, Nb, Nc = the NPC centers of A'A1A2, B'B1B2, C'C1C2, resp.

1. A1A2, B1B2, C1C2 are concurrent.

2. ABC, NaNbNc are orthologic

3. A'B'C', NaNbNc are orthologic.

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

 

[Peter Moses]:

 

 

Hi Antreas,

1)  

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

 

9 X[373] - 5 X[1843], 5 X[6] - 3 X[3060], 9 X[373] - 10 X[3589], 5 X[141] - 6 X[3819], 2 X[3631] - 3 X[3917], 3 X[51] - 4 X[6329], 3 X[5085] - X[6403], 3 X[597] - 2 X[9969], 5 X[3618] - 3 X[9971], 5 X[9822] - 6 X[10219], 5 X[3763] - 3 X[11188], 3 X[3819] - 5 X[11574], 5 X[182] - 4 X[12006], 5 X[9967] - X[12162], 3 X[3060] + 5 X[12220], 3 X[599] - X[12272], 9 X[3819] - 5 X[14913], 3 X[141] - 2 X[14913], 3 X[11574] - X[14913], X[6144] - 3 X[15531], 6 X[373] - 5 X[16776], 2 X[1843] - 3 X[16776], 4 X[3589] - 3 X[16776].

= on lines {{2,9973},{3,9972},{6,22},{51, 6329},{52,12007},{67,69},{141, 1368},{159,8780},{182,1658},{2 06,6593},{373,468},{376,974},{ 511,550},{524,3313},{597,9969} ,{599,12272},{888,13232},{1350 ,15073},{1503,9967},{3098,9976 },{3618,9971},{3630,8681},{ 3631,3917},{3763,11188},{5085, 6403},{5092,11649},{5097,10263 },{5157,8541},{5965,6101},{ 6144,15531},{7493,11746},{7502 ,12236},{7712,12824},{7804, 16983},{9822,10219},{10168, 16532},{10539,15580},{12283, 15069},{13201,16176}}.
= complement X(9973).
= midpoint of X(i) and X(j) for these {i,j}: {{6, 12220}, {1350, 15073}, {3313, 6467}, {12283, 15069}, {13201, 16176}}.
= reflection of X(i) in X(j) for these {i,j}: {{52, 12007}, {141, 11574}, {1843, 3589}, {10263, 5097}, {16983, 7804}}.
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1176, 11416, 6), (1843, 3589, 16776).

2)
(ABC, NaNbNc): 

 

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

on the Jerabek hyperbola and the lines {{265,13201},{2979,3519},{3521 ,15305},{6241,14861},{9833,112 70},{11457,15108},{13432,16266 }}.
.
= X(14864)-cross conjugate of X(4).

(NaNbNc,  ABC): 

 

2 a^10-4 a^8 b^2+a^6 b^4+a^4 b^6+a^2 b^8-b^10-4 a^8 c^2-8 a^6 b^2 c^2+9 a^4 b^4 c^2+3 b^8 c^2+a^6 c^4+9 a^4 b^2 c^4-2 a^2 b^4 c^4-2 b^6 c^4+a^4 c^6-2 b^4 c^6+a^2 c^8+3 b^2 c^8-c^10:: 

 

3 X[5892] - X[7553], X[1216] - 3 X[7667], 3 X[376] + X[11750], 9 X[376] - X[12278], 3 X[11750] + X[12278], 3 X[549] - X[13419], X[12225] + 3 X[14855], 3 X[10170] - X[16655], X[11819] - 3 X[16836].

on lines {{3,2918},{4,8717},{30,5462},{ 113,8718},{125,13564},{376,117 50},{382,15805},{539,15644},{5 41,10575},{542,10627},{548,669 6},{549,13419},{550,10264},{12 16,7667},{1503,5447},{1657,128 97},{5448,14791},{5892,7553},{ 6030,6143},{6699,7488},{7391, 13336},{7525,15578},{7691, 16003},{9530,15912},{10116,106 25},{10170,16655},{10539,16063 },{11585,12900},{11818,13347}, {11819,16836},{12103,13470},{ 12140,17506},{12225,14855},{ 12362,14915},{12605,14641},{ 13403,15704},{14810,14864}}.
= midpoint of X(i) and X(j) for these {i,j}: {{1657, 12897}, {10116, 10625}, {12103, 13470}, {12605, 14641}, {13403, 15704}}.

3)
(A'B'C',  NaNbNc): 

 

a^2 (2 a^6-5 a^4 b^2+4 a^2 b^4-b^6-5 a^4 c^2-2 a^2 b^2 c^2+b^4 c^2+4 a^2 c^4+b^2 c^4-c^6) (a^8-2 a^6 b^2+2 a^2 b^6-b^8-2 a^6 c^2-2 a^4 b^2 c^2+a^2 b^4 c^2+3 b^6 c^2+a^2 b^2 c^4-4 b^4 c^4+2 a^2 c^6+3 b^2 c^6-c^8):: 

 

on the Jerabek hyperbola of the medial triangle and the lines {{51,1493},{1147,3060},{1511,1 0625},{4550,6241},{10539,11423 }}.

(NaNbNc, A'B'C'): as (NaNbNc, ABC).

4) X(10691)

Best regards,
Peter.