HYACINTHOS 27743

[Antreas P. Hatzipolakis]:

Let ABC be a triangle and P a point.

Denote:

Na, Nb, Nc = the NPC centers of PBC, PCA, PAB, resp.

N1, N2, N3 = the NPC centers of PNbNc, PNcNa, PNaNb, resp.

For P = N:

NaNbNc, N1N2N3 are perspective at N.

1. The reflections of NaNN1, NbNN2, NcNN3 in the sidelines of NaNbNc : NbNc, NcNa, NaNb, resp. are concurrent.

Point of concurrence?

 

2.  Let A'B'C' be the pedal triangle of N.

Denote:

La, Lb, Lc = the reflections of NNa, NNb, NNc in NbNc, NcNa, NaNb, resp.

The parallels to La, Lb, Lc through A', B', C', resp. are concurrent.

Point of concurrence?


[Peter Moses]:


Hi Antreas,

1). X(14051).

2). (a^2-b^2-b c-c^2) (a^2-b^2+b c-c^2) (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (a^4-a^2 b^2+b^4-2 a^2 c^2-2 b^2 c^2+c^4) (a^4-2 a^2 b^2+b^4-a^2 c^2-2 b^2 c^2+c^4) (3 a^12-12 a^10 b^2+19 a^8 b^4-16 a^6 b^6+9 a^4 b^8-4 a^2 b^10+b^12-12 a^10 c^2+20 a^8 b^2 c^2-4 a^6 b^4 c^2-10 a^4 b^6 c^2+12 a^2 b^8 c^2-6 b^10 c^2+19 a^8 c^4-4 a^6 b^2 c^4+5 a^4 b^4 c^4-8 a^2 b^6 c^4+15 b^8 c^4-16 a^6 c^6-10 a^4 b^2 c^6-8 a^2 b^4 c^6-20 b^6 c^6+9 a^4 c^8+12 a^2 b^2 c^8+15 b^4 c^8-4 a^2 c^10-6 b^2 c^10+c^12)::
on lines {{5,930},{128,1154},{10615,14071}}.

Best regards,
Peter Moses.