HYACINTHOS 23083

Antreas Hatzipolakis

 

Let ABC be a triangle, P a point and A'B'C' the pedal triangle of P.

Denote:

Ab, Ac = the reflections of A in B', C', resp.

Na = the NPC center of AAbAc. Similarly Nb, Nc.

Which is the locus of P such that the NPC center of NaNbNc

is lying on the OP line?

I think the incenter I lies on the locus. Which is this point (= NPC center of NaNbNc) on the OI line?

Note: The NPCs  (Na), (Nb), (Nc) are concurrent for any points A', B', C'

on BC, CA, AB, resp. (see Hyacinthos #13218)

APH

 

[César Lozada]:

 

[APH]

> Which is the locus of P such that the NPC center of NaNbNc is lying on the OP line?

Locus=Stammler hyperbola, through excenters and ETC´s: 1, 3, 6, 155, 159, 195, 399, 610, 1498, 1740, 2574, 2575, 2916, 2917, 2918, 2929, 2930, 2931, 2935, 2948, 3216, 3360, 3499, 3511, 5898

I think the incenter I lies on the locus. Which is this point (N*= NPC center of NaNbNc) on the OI line?

N*(I)= 2*cos((B-C)/2)*sin(3*A/2)+2*(1-cos(A))*cos(B-C)-1 : : (trilinears)

     = Midpoint of: (5,3874), (3868,5694)                                                              

     = Reflection of: (3678/3628), (5885/942)    
     = On lines: (1,3), (5,3874), (355,3873), (546,2801), (758,5901), (912,5448), (946,1484), (952,3881), (1393,5399), (1483,3892), (1656,5904), (3628,3678), (3754,5844), (3868,5694), (4430,5818), (5690,5883)

     = (5*R+2*r)*X(1)-(R+2*r)*X(3)

    = ( -0.162996184703785, 0.15998637610432, 3.605133691390825 )

 

Other ETC-pairs (P,N*(P)) such that N*(P) lies on line (OP):

(3,5), (6,5097), (195,195), (399,110), (2574,2574), (2575,2575)

Note: X(2574) and X(2575) are intersections of Stammler hyperbola with line-at-infinity

 

Regards

César Lozada