HYACINTHOS 29238

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle and P a point.

Denote:

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

A', B', C' = the reflections of Na, Nb, Nc in BC, CA, AB, resp.
A", B", C" = the reflections of Na, Nb, Nc in  AP, BP, CP, resp.

Which is the locius of P such that A'A", B'B", C'C" are concurrent?

I lies on the locus

 

[César Lozada]:

 

Locus = {Linf} ∪ {excentral-circum-degree-16 through ETC’s X(1)}

 

For P=X(1), the point of concurrence is

 

Q( X(1) ) = X(500)X(1064) ∩ X(517)X(13630)

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

= lies on these lines: {500, 1064}, {517, 13630}

= [ 0.4943980419473895, 1.4811091259533970, 2.3870975291947710 ]

 

César Lozada