HYACINTHOS 29246

[Antreas P. Hatzipolakis]:

 Let ABC be a triangle.

Denote:

Na, Nb, Nc = the NPC centers of IBC, ICA, IAB, resp.

(Oa), (Ob), (Oc) = the circles (I, INa), (I, INb), (I, INc), resp.

(O1), (O2), (O3) = the reflections of (Oa), (Ob), (Oc) in BC, CA, AB, resp.
The radical axes of the pairs of the circles are parallels to the bisectors AI, BI, CI..
Their intersection (radical center of the circles) lies on the OI line..

Point?


[César Lozada]:

 
Q = X(1)X(3) ∩ X(5)X(33337)

 

[....]

 

 

[APH]:

In general:

Let ABC be a triangle and P a point.

Denote:

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

(Oa), (Ob), (Oc) = the circles (P, PNa), (P, PNb), (P, PNc), resp.

(O1), (O2), (O3) = the reflections of (Oa), (Ob), (Oc) in BC, CA, AB, resp.  

Probably the radical center of (O1), (O2), (O3) is interesting for other than P = I points.

Note 1:
We may ask for the locus of P such that the radical center lies on the OI or OH, or..... lines.

Now, in the case P = I, being the radical axes parallels to the corresponding angle bisectors AI, BI, CI, their reflections in these bisectors are concurrent at a point Q' = reflection of the above point Q in I.
Is it interesting?

Note 2
We may ask which is the locus of P such that the radical axes of ((O2), (O3)),  ((O3), (O1)) , ((O1), (O2)) in AP, BP, CP, resp. are concurrent.



 [César Lozada]:

The radical center Q* of (O1), (O2), (O3) is complicated (degree-9), then the loci for P on OI, OH, etc, are not easy to deal to,

 

Q’ = X(1)X(3) ∩ X(1483)X(21630)

= a*( 4*a^6-9*(b+c)*a^5-(3*b^2-28*b*c+3*c^2)*a^4+3*(3*b-2*c)*(2*b-3*c)*(b+c)*a^3-2*(3*b^4+3*c^4+(9*b^2-25*b*c+9*c^2)*b*c)*a^2-3*(b^2-c^2)*(b-c)*(3*b^2-7*b*c+3*c^2)*a+5*(b^2-c^2)^2*(b-c)^2) : : (barys)

= 5*X(1)-X(26285), 3*X(1)-X(26287), X(10525)+7*X(20057)

= lies on these lines: {1, 3}, {1483, 21630}, {5901, 6702}, {10525, 20057}, {28204, 32905}

= midpoint of X(24680) and X(26087)

= reflection of X(33657) in X(1)

= [ 3.2099684899026720, 2.9451314808623530, 0.1202033844322863 ]

 

The locus of P such that the reflections of the radical axes of {(O2), (O3)}, .. in (AP),… are concurrent is a degree-15 excentral-circumcurve. X(1) is the only ETC center on it.

 

César Lozada.