HYACINTHOS 21971

Antreas P. Hatzipolakis

A CIRCLE:

Let ABC be a triangle, A'B'C' the cevian triangle of I and N1, N2, N3
the NPC centers of IB'C', IC'A', IA'B', resp.

The points I, N1,N2,N3 are concyclic.

Center of the circle?

LOCUS:

Let ABC be a triangle, P a point, A'B'C' the cevian triangle of P and N1,
N2, N3
the NPC centers of PB'C', PC'A', PA'B', resp.

Which is the locus of P such that the points P, N1,N2,N3 are concyclic ?

APH

 

[César Lozada]:

 

Yes, they are concyclic.



The center X of the circle has trilinears:

2*cos(A)+4*sin(3*A/2)*cos(B/2-C/2)+ cos(B-C)+2 : :



ETC search: 2.387773069046934.., 2.38593313995937.., 0.886815506990847..



X=Midpoint of X(I),X(J) for these (I,J):

(1,500)




X lies on line X(I),X(J) for these (I,J):

(1,30), (3,81), (5,581), (21,323), (30,79), (58,5428), (79,500),
(140,3216),

(186,2906), (237,1896), (285,1082), (358,1882), (386,549), (500,554),
(511,1385),

(550,991), (554,1081), (1036,4022), (1081,1464), (1154,2646), (1464,1717),


(1675,4280), (1717,1836), (1836,3058), (2071,4309), (2072,3746),
(2771,3743),

(2943,4341), (3058,3649), (3108,3784), (3649,3655), (3655,3656),
(3656,3782),

(3704,3990), (3782,4654), (4654,4854), (4854,5160), (5160,5434),
(5434,5441)



Regards

César Lozada