HYACINTHOS 26727

 [Antreas P. Hatzipolakis]:

 

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

Denote

Ba, Ca = the orthogonal projections of B', C' on BC, resp.
(Nba), (Nca) = the NPCs of BaB'A', CaC'A', resp.

 

Cb, Ab = the orthogonal projections of C', A' on CA, resp.
(Ncb), (Nab) = the NPCs of CbC'B', AbA'B', resp.

 

Ac, Bc = the orthogonal projections of A', B' on AB, resp.
(Nac), (Nbc) = the NPCs of AcA'C', BcB'C', resp.

 

R1 = the radical axis of (Nba), (Nca)
R2 = the radical axis of (Ncb), (Nab)
R3 = the radical axis of (Nac), (Nbc)

1. R1, R2, R3 are concurrent at the NPC center N' of A'B'C'

 

S1 = the radical axis of (Nbc), (Ncb)
S2 = the radical axis of (Nca), (Nac)
S3 = the radical axis of (Nab), (Nba)

 

2. S1, S2, S3 are concurrent
They are perpendiculars to BC,CA, AB, resp. and also to NbcNcb, NcaNac, NabNba, resp, therefore the triangle A"B"C" bounded by NbcNcb, NcaNac, NabNba is homothetic with ABC.

3. Homothetic center of ABC, A"B"C" ?

T1 = the radical axis of (Nab), (Nac)
T2 = the radical axis of (Nbc), (Nba)
T3 = the radical axis of (Nca), (Nac)

 

A*B*C* = the triangle bounded by T1,T2,T3

4. ABC, A*B*C* are parallelogic.
The parallelogic center (ABC, A*B*C*) is the O.
The other one (A*B*C*,ABC) ?

[Peter Moses]:

Hi Antreas,

1.  X(143).

2.  X(389).

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

on lines {{2,576},{4,1173},{6,25},{22,575},{32,8565},{52,7514},{61,3132},{62,3131},{125,8889},{143,569},{182,3060},{185,1597},{186,578},{237,7772},{263,3108},{275,3168},{323,11451},{343,11548},{373,394},{378,389},{418,5158},{427,11470},{428,8550},{511,5422},{542,7394},{567,13321},{597,7499},{612,8540},{1092,5462},{1147,14627},{1181,3527},{1199,6759},{1351,3917},{1899,7378},{1990,6755},{1992,7392},{1993,5097},{1994,5640},{3066,3167},{3088,11431},{3148,5007},{3155,6420},{3156,6419},{3284,6641},{3292,5020},{5012,11002},{5032,7398},{5058,8577},{5062,8576},{5102,5650},{5133,5476},{5191,14075},{5446,10984},{5475,8035},{5480,11245},{5621,13417},{5890,13596},{5899,11692},{5946,13352},{6353,8537},{6417,10132},{6418,10133},{6515,14561},{6676,8538},{6748,14569},{6776,7408},{7395,14531},{7484,11477},{7487,10619},{7494,11511},{7500,11179},{7505,12242},{7544,10112},{7592,10110},{9786,11410},{9818,14831},{10095,10539},{10151,12233},{10154,11255},{10263,13336},{10565,11416},{11225,11442},{11422,13595},{11426,13367},{14483,14491}}.
3 X[5422] - X[7485].
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 25, 13366), (6, 51, 184), (6, 9777, 51), (25, 13366, 184), (51, 13366, 25), (389, 11424, 1204), (1199, 9781, 6759), (1351, 10601, 3917), (1495, 11402, 184), (1993, 5943, 5651), (1994, 5640, 9306), (5097, 5943, 1993), (5480, 11245, 11550), (8035, 8036, 5475), (10982, 11432, 185).
isogonal of the isotomic of X(1656).
X(1656)-Ceva conjugate of X(10979).
X(i)-isoconjugate of X(j) for these (i,j): {{75, 13472}}.
crosspoint of X(i) and X(j) for these (i,j): {{6, 3527}}.
crosssum of X(i) and X(j) for these (i,j): {{2, 631}, {3, 5422}}.
barycentric product X(i)X(j) for these {i,j}: {{4, 10979}, {6, 1656}, {216, 4994}}.
barycentric quotient X(i)/X(j) for these {i,j}: {{32, 13472}, {1656, 76}, {4994, 276}, {10979, 69}}.

4.  a^2 (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (2 a^4-3 a^2 b^2+b^4-3 a^2 c^2-2 b^2 c^2+c^4) (a^12-7 a^10 b^2+20 a^8 b^4-30 a^6 b^6+25 a^4 b^8-11 a^2 b^10+2 b^12-7 a^10 c^2+25 a^8 b^2 c^2-22 a^6 b^4 c^2-14 a^4 b^6 c^2+29 a^2 b^8 c^2-11 b^10 c^2+20 a^8 c^4-22 a^6 b^2 c^4-2 a^4 b^4 c^4-18 a^2 b^6 c^4+22 b^8 c^4-30 a^6 c^6-14 a^4 b^2 c^6-18 a^2 b^4 c^6-26 b^6 c^6+25 a^4 c^8+29 a^2 b^2 c^8+22 b^4 c^8-11 a^2 c^10-11 b^2 c^10+2 c^12)::

 

Best regards,

Peter Moses.