HYACINTHOS 26386

[Le Viet An]:

Let ABC be a triangle and MaMbMc the pedal triangle of O.
The line OI intersects MbMc, McMa, MaMb at Ka, Kb, Kc, resp.
The perpendiculars to OA, OB, OC from Ka, Kb, Kc, resp. bound a triangle A'B'C'
Then the circumcircle of A'B'C' touches the incircle.

Which point is:
1. the center of the circle?
2. the touchpoint?


[Peter Moses]:

Hi Antreas,

1).
a^2 (a^11 b^3-5 a^9 b^5+10 a^7 b^7-10 a^5 b^9+5 a^3 b^11-a b^13-a^11 b^2 c-2 a^10 b^3 c+8 a^9 b^4 c+7 a^8 b^5 c-22 a^7 b^6 c-8 a^6 b^7 c+28 a^5 b^8 c+2 a^4 b^9 c-17 a^3 b^10 c+2 a^2 b^11 c+4 a b^12 c-b^13 c-a^11 b c^2+4 a^10 b^2 c^2-3 a^9 b^3 c^2-18 a^8 b^4 c^2+21 a^7 b^5 c^2+28 a^6 b^6 c^2-35 a^5 b^7 c^2-16 a^4 b^8 c^2+24 a^3 b^9 c^2-6 a b^11 c^2+2 b^12 c^2+a^11 c^3-2 a^10 b c^3-3 a^9 b^2 c^3+22 a^8 b^3 c^3-7 a^7 b^4 c^3-46 a^6 b^5 c^3+20 a^5 b^6 c^3+35 a^4 b^7 c^3-12 a^3 b^8 c^3-10 a^2 b^9 c^3+a b^10 c^3+b^11 c^3+8 a^9 b c^4-18 a^8 b^2 c^4-7 a^7 b^3 c^4+40 a^6 b^4 c^4+3 a^5 b^5 c^4-28 a^4 b^6 c^4-15 a^3 b^7 c^4+12 a^2 b^8 c^4+11 a b^9 c^4-6 b^10 c^4-5 a^9 c^5+7 a^8 b c^5+21 a^7 b^2 c^5-46 a^6 b^3 c^5+3 a^5 b^4 c^5+10 a^4 b^5 c^5+15 a^3 b^6 c^5+8 a^2 b^7 c^5-18 a b^8 c^5+5 b^9 c^5-22 a^7 b c^6+28 a^6 b^2 c^6+20 a^5 b^3 c^6-28 a^4 b^4 c^6+15 a^3 b^5 c^6-24 a^2 b^6 c^6+9 a b^7 c^6+4 b^8 c^6+10 a^7 c^7-8 a^6 b c^7-35 a^5 b^2 c^7+35 a^4 b^3 c^7-15 a^3 b^4 c^7+8 a^2 b^5 c^7+9 a b^6 c^7-10 b^7 c^7+28 a^5 b c^8-16 a^4 b^2 c^8-12 a^3 b^3 c^8+12 a^2 b^4 c^8-18 a b^5 c^8+4 b^6 c^8-10 a^5 c^9+2 a^4 b c^9+24 a^3 b^2 c^9-10 a^2 b^3 c^9+11 a b^4 c^9+5 b^5 c^9-17 a^3 b c^10+a b^3 c^10-6 b^4 c^10+5 a^3 c^11+2 a^2 b c^11-6 a b^2 c^11+b^3 c^11+4 a b c^12+2 b^2 c^12-a c^13-b c^13)::
on line {1, 3025}.

2).
X(3025).

Best regards,
Peter Moses.