HYACINTHOS 26438

[Antreas P. Hatzipolakis]:

Let ABC be a triangle.

Denote:

Na,Nb,Nc = the NPC centers of OBC, OCA, OAB, resp.

The Euler lines of NNbNc, NNcNa, NNaNb and NaNbNc are concurrent at a point U.

1. Which point is the N wrt triangle NaNbNc (lying on its Neuberg cubic) ?

 

2. Which point is the U wrt triangle NaNbNc ?

3. Which point is the U wrt triangle ABC ?

[César Lozada]:

 

 

1)      X(1) = I  (for acute angled ABC)

2)      X(21) (for acute angled ABC)

3)      U = complement of X(10274)

= cos(2*A)*cos(B-C)+cos(3*A)* cos(2*(B-C))-(cos(2*A)+1)*cos( 3*(B-C)) : : (trilinears)

= On lines: {2,10274}, {3,161}, {5,10628}, {54,125}, {427,11808}, {575,8254}, {1092,2888}, {1154,5449}, {3153,7691}, {3567,3574}, {5498,12038}, {5965,8548}

= complement of X(10274)

= [ 6.946659154843544, 0.33883952283847, 0.199932894629959 ]

 

César Lozada