HYACINTHOS 26377

 

[Le Viet An]:

Let ABC be a triangle and HaHbHc, TaTbTc the orthic, extouch triangles, resp.

The reflections of HbHc, HcHa,HaHb in TbTc, TcTa, TaTb, resp. bound a triangle A'B'C'.

Then the circumcircle of A'B'C' touches the incircle of ABC.

Which is

1. the center of the circle ?

2. the touchpoint ?


[Peter Moses]:


Hi Antreas,

1).
a (a^12-6 a^10 b^2+15 a^8 b^4-20 a^6 b^6+15 a^4 b^8-6 a^2 b^10+b^12+4 a^10 b c+4 a^9 b^2 c-8 a^8 b^3 c-16 a^7 b^4 c+24 a^5 b^6 c+8 a^4 b^7 c-16 a^3 b^8 c-4 a^2 b^9 c+4 a b^10 c-6 a^10 c^2+4 a^9 b c^2+2 a^8 b^2 c^2+20 a^6 b^4 c^2-8 a^5 b^5 c^2-28 a^4 b^6 c^2+18 a^2 b^8 c^2+4 a b^9 c^2-6 b^10 c^2-8 a^8 b c^3+16 a^6 b^3 c^3-16 a^5 b^4 c^3-8 a^4 b^5 c^3+32 a^3 b^6 c^3-16 a b^8 c^3+15 a^8 c^4-16 a^7 b c^4+20 a^6 b^2 c^4-16 a^5 b^3 c^4+26 a^4 b^4 c^4-16 a^3 b^5 c^4-12 a^2 b^6 c^4-16 a b^7 c^4+15 b^8 c^4-8 a^5 b^2 c^5-8 a^4 b^3 c^5-16 a^3 b^4 c^5+8 a^2 b^5 c^5+24 a b^6 c^5-20 a^6 c^6+24 a^5 b c^6-28 a^4 b^2 c^6+32 a^3 b^3 c^6-12 a^2 b^4 c^6+24 a b^5 c^6-20 b^6 c^6+8 a^4 b c^7-16 a b^4 c^7+15 a^4 c^8-16 a^3 b c^8+18 a^2 b^2 c^8-16 a b^3 c^8+15 b^4 c^8-4 a^2 b c^9+4 a b^2 c^9-6 a^2 c^10+4 a b c^10-6 b^2 c^10+c^12)::
on the lines {1, 1361}.

2).
X(1364).

Best regards,
Peter Moses.