HYACINTHOS 27777

[Antreas P. Hatzipolakis]:

Let ABC be a triangle.

Denote:

A',B',C' = the reflections of H in BC,CA,AB, resp.

N1, N2, N3 = the isogonal conjugates of N wrt triangles A'BC, B'CA, C'AB, resp.

ABC, N1N2N3 are cyclologic.

Cyclologic centers?

[Peter Moses]:

Hi Antreas,
 
(ABC, N1N2N3):

(a^4-2 a^2 b^2+b^4-a^2 c^2-b^2 c^2) (a^4-a^2 b^2-2 a^2 c^2-b^2 c^2+c^4) (a^8+a^6 b^2-4 a^4 b^4+a^2 b^6+b^8+a^6 c^2+5 a^4 b^2 c^2-a^2 b^4 c^2-4 b^6 c^2-4 a^4 c^4-a^2 b^2 c^4+6 b^4 c^4+a^2 c^6-4 b^2 c^6+c^8)::

on lines {{30,54},{93,186},{476,1141},{477,933},{1173,16106},{5899,16035},{15646,19176}}.
crosssum of X(1154) and X(14128).
barycentric product X(95) {{6,382},{50,112},{93,393},...}.

and

(N1N2N3, ABC):

X(54).

Best regards,
Peter Moses.