HYACINTHOS 29576

 

[Kadir Altintas]

Let ABC be a triangle with H = Orthocenter and N = the NPC center and A'B'C' the pedal triangle of N

Denote:

Ha, Hb, Hc = the orthocenters of NBC, NCA, NAB, resp.
Na, Nb, Nc = the NPC centers of NHbHc, NHcHa, NHaHb, resp. 

Prove:  
1. Na, Nb, Nc and H are concyclic.
2. A'B'C', NaNbNc are perspective.

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[Ercole Suppa]

Let X be the perspector ofA'B'C' and NaNbNc. 

 

We have:

X = X(5)X(25043) ∩ X(930)X(30480)

= b^2 c^2 (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) (-a^2 b^2+b^4-a^2 c^2-2 b^2 c^2+c^4) (-2 a^12+7 a^10 b^2-8 a^8 b^4+2 a^6 b^6+2 a^4 b^8-a^2 b^10+7 a^10 c^2-8 a^8 b^2 c^2-3 a^6 b^4 c^2+3 a^4 b^6 c^2+b^10 c^2-8 a^8 c^4-3 a^6 b^2 c^4-4 a^4 b^4 c^4+a^2 b^6 c^4-4 b^8 c^4+2 a^6 c^6+3 a^4 b^2 c^6+a^2 b^4 c^6+6 b^6 c^6+2 a^4 c^8-4 b^4 c^8-a^2 c^10+b^2 c^10) : : (barys)

= (SC^2-3 S^2) (SW-SB) (SW-SC) (S^2+SB SC) (-4 S^2-SB SC+SB SW-SC^2+SC SW) (-2 R^4+10 R^2 SB+10 R^2 SC-2 R^2 SW+S^2-4 SB SW-4 SC SW+SW^2) : : (barys)

= lies on the cubic K054 and these lines: {5,25043}, {930,30480}

= (6-9-13) search numbers [0.1492354117254041002, -0.4684044961106092614, 3.8960666353415374686]


Best regards,
Ercole Suppa