HYACINTHOS 29617

[Kadir Atintas]:

 

Let ABC be a triangle and P be a point. 

 

Denote:

 

Oa, Ob, Oc = the circumcenters of PBC, PCA, PAB, resp. 

Oaa, Obb, Occ = the  circumcenters of PObOc, POcOa, POaOb, resp. 

 

Which is the locus of P such that OaObOc, OaaObbOcc are perspective?

 

The Neuberg cubic  + + ? 

Some perspectors?

 

 

[César Lozada]:

 

 

Locus = {Linf} ∪ {K001: Neuberg cubic}

 

ETC-pairs {P, Q(P)=perspector}: {1, 1385}, {3, 3}, {4, 6288}, {13, 2}, {14, 2}, {30, 30}, {74, 3}, {1263, 54}, {3065, 22936}

 

 

Q( X(15) ) = X(3)X(54) ∩ X(15)X(1337)

= a^2*(-4*(-a^2+b^2+c^2)*S+(a^4-(b^2+c^2)*a^2-b^2*c^2)*sqrt(3)) : :

= lies on these lines: {2, 16626}, {3, 54}, {15, 1337}, {23, 21401}, {61, 21461}, {110, 11146}, {396, 11139}, {3060, 22236}, {3129, 14170}, {3132, 12834}, {3171, 10645}, {5238, 6030}, {5888, 11130}, {8929, 16962}, {11480, 15080}, {14704, 31940}

= [ 6.1629850426573940, 6.1270086602451980, -3.4455653794887880 ]

 

Q( X(16) ) = X(3)X(54) ∩ X(16)X(1338)

= a^2*(4*(-a^2+b^2+c^2)*S+(a^4-(b^2+c^2)*a^2-b^2*c^2)*sqrt(3)) : :

= lies on these lines: {2, 16627}, {3, 54}, {16, 1338}, {23, 21402}, {62, 21462}, {110, 11145}, {395, 11138}, {3060, 22238}, {3130, 14169}, {3131, 12834}, {3170, 10646}, {5237, 6030}, {5888, 11131}, {8930, 16963}, {11481, 15080}, {14705, 31939}

= [ -5.4115324990918910, 10.4635478189837600, -1.1056997778081180 ]

 

César Lozada