HYACINTHOS 25666

[Antreas P. Hatzipolaks]:

 

Let ABC be a triangle and IaIbIc the antipedal triangle of I.

Denote:

Na, Nb, Nc = the NPC centers of IaBC, IbCA, IcAB, resp.

Naa, Nab, Nac = the reflections of Na in IA,IB,IC, resp.
Nba, Nbb, Nbc = the reflections of Nb in IA,IB,IC, resp.

Nca, Ncb, Ncc = the reflections of Nc in IA,IB,IC, resp.

N1, N2, N3 = the NPC centers of NaaNabNac, NbaNbbNbc, NcaNcbNcc, resp.

 

R1, R2, R3 = the Euler lines of NaaNabNac, NbaNbbNbc, NcaNcbNcc, resp. (conccurrent at their common circumcenter I)

1. The parallels to R1, R, R3 through A, B, C, resp. are concurrent.
2. The parallels to R1, R, R3 through Ia, Ib, Ic, resp. are concurrent.

3. ABC, N1N2N3 are orthologic.

4. IaIbIc, N1N2N3 are orthologic

5. The orthocenter of N1N2N3 lies on the Euler line of ABC (?).

[Peter Moses]:

Hi Antreas,

 

1). a (a^3-3 a^2 b-a b^2+3 b^3-a^2 c+5 a b c-b^2 c-a c^2-3 b c^2+c^3) (a^3-a^2 b-a b^2+b^3-3 a^2 c+5 a b c-3 b^2 c-a c^2-b c^2+3 c^3)::
on lines {{1,6797},{3,11279},{4,9897},{ 8,11524},{11,5559},{21,2802},{ 79,952},{80,5844},{104,484},{ 498,7320},{517,3065},{519, 11604},{528,3255},...}
on Feuerbach, Reflection of X(5559) in X(11).

 

2). a (a^6-2 a^5 b-a^4 b^2+4 a^3 b^3-a^2 b^4-2 a b^5+b^6-2 a^5 c+12 a^4 b c-22 a^3 b^2 c-6 a^2 b^3 c+24 a b^4 c-6 b^5 c-a^4 c^2-22 a^3 b c^2+67 a^2 b^2 c^2-34 a b^3 c^2-b^4 c^2+4 a^3 c^3-6 a^2 b c^3-34 a b^2 c^3+12 b^3 c^3-a^2 c^4+24 a b c^4-b^2 c^4-2 a c^5-6 b c^5+c^6)::
on lines {{40,7993},{191,2802},...}.
On the Jerabek of the excentral triangle.

 

3a). X(3065).

3b). N1N2N3, ABC: a (a^5 b-a^4 b^2-2 a^3 b^3+2 a^2 b^4+a b^5-b^6+a^5 c+3 a^3 b^2 c-2 a^2 b^3 c-4 a b^4 c+2 b^5 c-a^4 c^2+3 a^3 b c^2-6 a^2 b^2 c^2+3 a b^3 c^2+b^4 c^2-2 a^3 c^3-2 a^2 b c^3+3 a b^2 c^3-4 b^3 c^3+2 a^2 c^4-4 a b c^4+b^2 c^4+a c^5+2 b c^5-c^6)::
on lines {{1,3},{10,2771},{30,3754},{ 140,2800},{355,6951},{500, 4642},{549,3878},,...}.
Midpoint of X(i) and X(j) for these {i,j}: {{65, 3579},{5690,5884}}.
Reflection of X(i) in X(j) for these {i,j}: {{6583, 5885},{9955,3812}}.
3 X[1385] - X[3057], 2 X[942] - 3 X[5885], 3 X[3] + X[5903], 4 X[942] - 3 X[6583], 3 X[10202] - X[10222], 3 X[10246] - X[10284], 3 X[354] - X[11278], 3 X[354] - 4 X[12009], X[11278] - 4 X[12009].
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,484,3579),(1385,3579,35).

 

4a). IaIbIc, N1N2N3: a (a^6-2 a^5 b-a^4 b^2+4 a^3 b^3-a^2 b^4-2 a b^5+b^6-2 a^5 c-4 a^4 b c+2 a^3 b^2 c+2 a^2 b^3 c+2 b^5 c-a^4 c^2+2 a^3 b c^2+3 a^2 b^2 c^2-2 a b^3 c^2-b^4 c^2+4 a^3 c^3+2 a^2 b c^3-2 a b^2 c^3-4 b^3 c^3-a^2 c^4-b^2 c^4-2 a c^5+2 b c^5+c^6)::
on lines {{1,149},{9,1030},{21,5506},{ 30,5538},{40,2771},{80,3925},{ 100,191},{214,5284},...}.
Reflection of X(i) in X(j) for these {i,j}: {{149, 11263}, {191, 100},{1768,3651}}.
On the Jerabek of the excentral triangle.

4b). X(10).

 

5). X(2475) ... on ABC’s Euler.

 

Best regards,

Peter Moses.