ADGEOM 1525

[Tran Quang Hung, ADGEOM #1506]:

 

Let ABC be a triangle with Feruerbach point Fa,Fb,Fc and excircles (Ia),(Ib),(Ic).

 

(Ka) is the circle passing through Fb,Fc and touches (Ia) at D but (Ka) is not NPC.

(Kb) is the circle passing through Fc,Fa and touches (Ib) at E but (Kb) is not NPC.

(Kc) is the circle passing through Fa,Fb and touches (Ic) at F but (Kb) is not NPC.

 

I can't draw the circle touch (Ka),(Kb),(Kc). I conjecture that the if (K) is the circle which touches (Ka),(Kb),(Kc) and (L) is the circle touches (Ia),(Ib),(Ic) internally then (K) and (L) are tangent.Fa,Fb and touches (Ic) at F but (Kb) is not NPC.

 

 

[Peter Moses,  ADGEOM #1508]:

 

Apollonius circle  and (K) are tangent at 

a^2 (a b+b^2+a c+c^2)^2 (a^3+b^3+a b c-2 b c^2-c^3) (a^3-b^3+a b c-2 b^2 c+c^3)::

(is now X(5975) in ETC).

 

 

[Angel Montesdeoca]:

 

*** In addition:

 

The radical axes of Apollonius circle and each circles (Ka), (Kb), (Kc) bound a triangle which is perspective with  ABC. The perspector is

(a(b+c)^2(a(b+c)+b^2+c^2)^2:...:...),

with (6-9-13)-search number 2.86232065098383724957358

 

Best regards,

Angel Montesdeoca