HYACINTHOS 25017

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle and A'B'C', A"B"C" the pedal triangles of H, N, resp.

Denote:

A*, B*, C* = the reflections of N in BC, CA, AB, resp.

Aa, Ab, Ac = the orthogonal projections of A* on AA', BB', CC', resp.
Ba, Bb, Bc = the orthogonal projections of B* on AA', BB', CC', resp.
Ca, Cb, Cc = the orthogonal projections of C* on AA', BB', CC', resp.

La, Lb, Lc = the Euler lines of AaAbAc, BaBbBc, CaCbCc, resp.
L1, L2, L3 = the reflections of La, Lb, Lc in BC, CA, AB, resp.

 

1. La, Lb, Lc are concurrent

2. L1, L2, L3, are concurrent

3. the reflections of La, Lb, Lc in AA', BB', CC', resp. are concurrent

4. the parallels to La, Lb, Lc through A, B, C resp. are concurrent.
5. the parallels to La, Lb, Lc through A', B', C' resp. are concurrent
6. the parallels to La, Lb, Lc through A", B", C" resp. are concurrent
7. the parallels to La, Lb, Lc through A*, B*, C* resp. are concurrent

8. the parallels to L1, L2, L3 through A', B', C', resp. are concurrent.

 

[Peter Moses]:

Hi Antreas,

 

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

on lines {{2,7731},{5,10628},...}.

2) X(10096).

3) 2 a^10-a^8 b^2-6 a^6 b^4+4 a^4 b^6+4 a^2 b^8-3 b^10-a^8 c^2+18 a^6 b^2 c^2-7 a^4 b^4 c^2-19 a^2 b^6 c^2+9 b^8 c^2-6 a^6 c^4-7 a^4 b^2 c^4+30 a^2 b^4 c^4-6 b^6 c^4+4 a^4 c^6-19 a^2 b^2 c^6-6 b^4 c^6+4 a^2 c^8+9 b^2 c^8-3 c^10::
on Euler line {2,3}

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

On lnes {{4,10264},{54,5663},{64,9919} ,{74,2070},{125,3521},...}.

5) {a^2 (a^8 b^2-2 a^6 b^4+2 a^2 b^8-b^10+a^8 c^2+4 a^6 b^2 c^2-a^4 b^4 c^2-7 a^2 b^6 c^2+3 b^8 c^2-2 a^6 c^4-a^4 b^2 c^4+10 a^2 b^4 c^4-2 b^6 c^4-7 a^2 b^2 c^6-2 b^4 c^6+2 a^2 c^8+3 b^2 c^8-c^10) (a^10-3 a^8 b^2+2 a^6 b^4+2 a^4 b^6-3 a^2 b^8+b^10-3 a^8 c^2+7 a^6 b^2 c^2-4 a^4 b^4 c^2-a^2 b^6 c^2+b^8 c^2+2 a^6 c^4-4 a^4 b^2 c^4+6 a^2 b^4 c^4-2 b^6 c^4+2 a^4 c^6-a^2 b^2 c^6-2 b^4 c^6-3 a^2 c^8+b^2 c^8+c^10)::

6) a^2 (a^2-b^2-b c-c^2) (a^2-b^2+b c-c^2) (a^8 b^2-2 a^6 b^4+2 a^2 b^8-b^10+a^8 c^2+4 a^6 b^2 c^2-a^4 b^4 c^2-7 a^2 b^6 c^2+3 b^8 c^2-2 a^6 c^4-a^4 b^2 c^4+10 a^2 b^4 c^4-2 b^6 c^4-7 a^2 b^2 c^6-2 b^4 c^6+2 a^2 c^8+3 b^2 c^8-c^10)::

On lines {{3,7731},{5,113},{110,6102},. ..}.

7) a^2 (a^12 b^2-4 a^10 b^4+5 a^8 b^6-5 a^4 b^10+4 a^2 b^12-b^14+a^12 c^2-2 a^8 b^4 c^2-5 a^6 b^6 c^2+12 a^4 b^8 c^2-7 a^2 b^10 c^2+b^12 c^2-4 a^10 c^4-2 a^8 b^2 c^4+14 a^6 b^4 c^4-9 a^4 b^6 c^4-2 a^2 b^8 c^4+3 b^10 c^4+5 a^8 c^6-5 a^6 b^2 c^6-9 a^4 b^4 c^6+10 a^2 b^6 c^6-3 b^8 c^6+12 a^4 b^2 c^8-2 a^2 b^4 c^8-3 b^6 c^8-5 a^4 c^10-7 a^2 b^2 c^10+3 b^4 c^10+4 a^2 c^12+b^2 c^12-c^14)::

On lines {{3,8157},{5,113},{20,7731},{5 1,10113},{52,1986},{110,186},. ..}.

8) a^8 b^2-2 a^6 b^4+2 a^2 b^8-b^10+a^8 c^2+4 a^6 b^2 c^2-a^4 b^4 c^2-7 a^2 b^6 c^2+3 b^8 c^2-2 a^6 c^4-a^4 b^2 c^4+10 a^2 b^4 c^4-2 b^6 c^4-7 a^2 b^2 c^6-2 b^4 c^6+2 a^2 c^8+3 b^2 c^8-c^10::

On lines {{2,3},{113,1154},...}.

 

 

Best regards,

Peter Moses.