HYACINTHOS 26733

[Antreas P. Hatzipolakis]:

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

Denote:

 

A*B*C* = the medial triangle of A'B'C'

 

Aa, Bb, Cc = the orthogonal projections of A', B', C' on AI, BI, CI, resp.

 

Nab, Nac = the NPC centers of AaA'B*, AaA'C* 

Nbc, Nba = the NPC centers of BbB'C*, BbB'A* 

Nca, Ncb = the NPC centers of CcC'A*, CcC'B*

T1 = the triangle bounded by NabNac, NbcNba, NcaNcb
T2 = the triangle bounded by NbaNca, NcbNab, NacNbc
T3 = the triangle bounded by NbcNcb, NcaNac, NabNba

 

 

1. ABC, T1 

2. ABC, T2  
3. ABC, T3  

are orthologic

 

The perpendicular bisectors of:

4. NabNac, NbcNba, NcaNcb

5. NbaNca, NcbNab, NacNbc

6. NbcNcb,NcaNac, NabNba

are concurrent.

 

PS: Is the Euler line of T1 passing through I? If so, which is the I wrt triangle T1 ?


[Peter Moses]:


Hi Antreas,

1).
ABC T1: X(1).
T1 ABC: X(65).

2).
ABC T2: X(7).

T2 ABC: (a-b-c) (2 a^4-5 a^3 b+5 a^2 b^2-3 a b^3+b^4-5 a^3 c-10 a^2 b c+3 a b^2 c-4 b^3 c+5 a^2 c^2+3 a b c^2+6 b^2 c^2-3 a c^3-4 b c^3+c^4):: 

on lines {{7,9580},{8,9},{55,6666},{142,497},{515,6767},{516,942},{518,12575},{527,3058},{553,11025},{1001,4314},{1058,5732},{3174,3452},{3243,9785},{3911,7676},{3946,4319},{5542,12699},{5728,10624},{5745,8730},{5750,14942},{6284,12573},{7675,12053},{8232,10389},{8236,10106},{11019,11495}}.
midpoint of X(i) and X(j) for these {i,j}: {{390, 950}, {5728, 10624}, {6284, 12573}}.
3 X[8236] - X[10106], 3 X[553] - 5 X[11025], 3 X[3058] + X[14100].
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (390, 5809, 1697), (497, 4326, 142).

3).
ABC T3: X(5557).

T3 ABC: 2 a^7-2 a^6 b-3 a^5 b^2+3 a^4 b^3+a b^6-b^7-2 a^6 c-24 a^5 b c+17 a^4 b^2 c+16 a^3 b^3 c-16 a^2 b^4 c+8 a b^5 c+b^6 c-3 a^5 c^2+17 a^4 b c^2+88 a^3 b^2 c^2+16 a^2 b^3 c^2-a b^4 c^2+3 b^5 c^2+3 a^4 c^3+16 a^3 b c^3+16 a^2 b^2 c^3-16 a b^3 c^3-3 b^4 c^3-16 a^2 b c^4-a b^2 c^4-3 b^3 c^4+8 a b c^5+3 b^2 c^5+a c^6+b c^6-c^7:: 

on lines {{2,3303},{497,13865},{942,10624},{7354,11037}}.

4). X(942).

5). a (3 a^4 b-6 a^3 b^2+6 a b^4-3 b^5+3 a^4 c+6 a^3 b c-12 a^2 b^2 c+2 a b^3 c+b^4 c-6 a^3 c^2-12 a^2 b c^2-16 a b^2 c^2+2 b^3 c^2+2 a b c^3+2 b^2 c^3+6 a c^4+b c^4-3 c^5):: 

on lines {{1,5779},{11,11018},{390,517},{495,9947},{516,12433},{518,3635},{942,4312},{946,971},{2951,5708},{3243,4930},{3295,10398},{3579,4326},{5049,8581},{5223,6767},{5806,12564},{7675,13624}}.
midpoint of X(942) and X(14100).
reflection of X(5045) in X(5572).
3 X[942] - X[4312], 3 X[5045] - 2 X[5542], X[5542] - 3 X[5572], X[390] + 3 X[5728], X[390] - 9 X[7671], X[5728] + 3 X[7671], 5 X[5728] - X[7672], 15 X[7671] + X[7672], 5 X[390] + 3 X[7672], 11 X[390] - 3 X[7673], 11 X[5728] + X[7673], 11 X[7672] + 5 X[7673], 3 X[5049] - X[8581], X[4312] + 3 X[14100].

6). a (3 a^5 b-3 a^4 b^2-6 a^3 b^3+6 a^2 b^4+3 a b^5-3 b^6+3 a^5 c+14 a^4 b c-23 a^3 b^2 c-11 a^2 b^3 c+20 a b^4 c-3 b^5 c-3 a^4 c^2-23 a^3 b c^2-74 a^2 b^2 c^2-23 a b^3 c^2+3 b^4 c^2-6 a^3 c^3-11 a^2 b c^3-23 a b^2 c^3+6 b^3 c^3+6 a^2 c^4+20 a b c^4+3 b^2 c^4+3 a c^5-3 b c^5-3 c^6):: 

on lines {{1,748},{1479,5556}}.

Best regards,
Peter Moses.