HYACINTHOS 25080

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle, A'B'C' the pedal triangle of N and A"B"C" the pedal triangle of O.

Denote:

A*, B*, C* = the reflections of A", B", C" in B'C', C'A', A'B', resp.

The circumcircles of NA"A*, NB"B*, NC"C* are coaxial.

The other than N point of concurrence?

 

[Peter Moses]:

Hi Antreas,

 

a^2 (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (a^14 b^2-6 a^12 b^4+15 a^10 b^6-20 a^8 b^8+15 a^6 b^10-6 a^4 b^12+a^2 b^14+a^14 c^2-8 a^12 b^2 c^2+19 a^10 b^4 c^2-13 a^8 b^6 c^2-13 a^6 b^8 c^2+26 a^4 b^10 c^2-15 a^2 b^12 c^2+3 b^14 c^2-6 a^12 c^4+19 a^10 b^2 c^4-14 a^8 b^4 c^4-5 a^6 b^6 c^4-5 a^4 b^8 c^4+22 a^2 b^10 c^4-11 b^12 c^4+15 a^10 c^6-13 a^8 b^2 c^6-5 a^6 b^4 c^6+6 a^4 b^6 c^6-8 a^2 b^8 c^6+17 b^10 c^6-20 a^8 c^8-13 a^6 b^2 c^8-5 a^4 b^4 c^8-8 a^2 b^6 c^8-18 b^8 c^8+15 a^6 c^10+26 a^4 b^2 c^10+22 a^2 b^4 c^10+17 b^6 c^10-6 a^4 c^12-15 a^2 b^2 c^12-11 b^4 c^12+a^2 c^14+3 b^2 c^14)::
on line {5,51}.

 

Best regards,

Peter Moses.