Τετάρτη 23 Οκτωβρίου 2019

HYACINTHOS 25789

 
[Tran Quang Hung]:

 

Let ABC be a triangle with incenter I.
 
Ka,Kb,Kc are the centers of mixtiliner incircles of ABC at the vertex A,B,C, reps.
 
Ka',Kb',Kc' are isogonal conjugate points of Ka,Kb,Kc wrt ABC, reps.
 
Then isogonal conjugate of I wrt Ka'Kb'Kc' lies on OI line of ABC.
 
Which is this point ?
.

[Angel Montesdeoca]:


Dear Tran Quang Hung.

The isogonal conjugate of I wrt Ka'Kb'Kc is

(r - 4 R) R X(1) + 2 r (2 R - r) X(O) = 
(a^2 (a^5-a^4 (b+c)+a^3 (-2 b^2+7 b c-2 c^2)+a (b-c)^2 (b^2-5 b c+c^2)-(b-c)^2 (b^3+c^3)+a^2 (2 b^3-b^2 c-b c^2+2 c^3)) : ...: ...)

on lines {1,3}

With (6-9-13)-search numbers (0.302697058407055,  0.544521460664727,  3.12398175141324)

Angel Montesdeoca

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