HYACINTHOS 26162

[Antreas P. Hatzipolakis]:

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

Denote:

Ab, Ac =the orthogonal projections of A' on HB', HC', resp.

(N1) = the NPC of A'AbAc

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

1. The radical center of (Na), (Nb), (Nc) lies on the Euler line of ABC.

2. ABC, NaNbNc are orthologic.
The orthologic center (ABC, NaNbNc) is the O.
The orthologic center (NaNbNc, ABC) lies on the Euler line.

 

3. A'B'C', NaNbNc are perspective.
The perspector lies on the Euler line.

[Peter Moses]:

Hi Antreas,

 

1).

2 a^10+a^8 b^2-10 a^6 b^4+4 a^4 b^6+8 a^2 b^8-5 b^10+a^8 c^2+16 a^6 b^2 c^2+4 a^4 b^4 c^2-36 a^2 b^6 c^2+15 b^8 c^2-10 a^6 c^4+4 a^4 b^2 c^4+56 a^2 b^4 c^4-10 b^6 c^4+4 a^4 c^6-36 a^2 b^2 c^6-10 b^4 c^6+8 a^2 c^8+15 b^2 c^8-5 c^10:: 

on line {2,3}.

{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5,3845,12084),(381,7528,3858) ,(546,5066,10224),(546,6677,4) ,(3091,7392,3851).

 

 

2). (NaNbNc, ABC) orthology: 

(a^2+b^2-c^2) (a^2-b^2+c^2) (2 a^6-3 a^4 b^2+b^6-3 a^4 c^2+12 a^2 b^2 c^2-b^4 c^2-b^2 c^4+c^6):: 

on lines {{2,3},{6,5878},{53, 5065},{68,11472},{578,2883},{ 800,6748},{952,1902},{1503, 13403},{1876,12433},{1968, 5305},{1990,7765},{2777,10110} ,{3426,12324},{3564,12162},{ 5663,13292},{5894,11438},{ 6000,12241},{6146,11381},{ 6241,11245},{6776,12315},{ 7583,11473},{7584,11474},{ 8550,9968},{11433,12250},{ 11475,11542},{11476,11543},{ 12022,12290}}.

Midpoint of X(i) and X(j) for these {i,j}: {{4, 1885}, {382, 12605}, {6146, 11381}}.

Reflection of X(i) in X(j) for these {i,j}: {{6756, 4}}.

5 X[4] - 3 X[428], 3 X[428] + 5 X[1885], 9 X[428] - 5 X[3575], 3 X[4] - X[3575], 3 X[1885] + X[3575], 5 X[3575] - 3 X[6240], 5 X[4] - X[6240], 3 X[428] - X[6240], 5 X[1885] + X[6240], 6 X[428] - 5 X[6756], 2 X[6240] - 5 X[6756], 2 X[3575] - 3 X[6756], 2 X[1885] + X[6756], 3 X[3830] - X[7553], 7 X[6240] - 15 X[7576], 7 X[3575] - 9 X[7576], 7 X[6756] - 6 X[7576], 7 X[428] - 5 X[7576], 7 X[4] - 3 X[7576], 7 X[1885] + 3 X[7576], X[3529] - 3 X[7667], 5 X[3] - 6 X[7734], 3 X[381] - 2 X[9825], 4 X[3850] - 3 X[10127], 7 X[3851] - 6 X[10128], 2 X[550] - 3 X[10691], X[6241] - 3 X[11245], 3 X[3543] + X[12225], 3 X[12022] + X[12290], 13 X[5079] - 12 X[13361].

{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,4,1596),(4,20,1598),(4,24, 1906),(4,378,235),(4,427,546), (4,1593,5),(4,1594,10151),(4, 1597,1595),(4,3088,381),(4, 3529,6995),(4,6240,428),(4, 7507,3845),(20,6804,3),(235, 378,140),(381,3548,5),(468, 3520,3530),(1594,10151,3850),( 3516,3542,549),(7577,10019, 12811).

 

3).

X(1594).

 

Best regards,

Peter Moses.