ADGEOM 1503 - ADGEOM 1508

#1503

 

Dear geometers,

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.

Then tangent at D,E,F of (Ia),(Ib),(Ic), reps bound a triangle which is perspective with triangle FaFbFc.

This perspector is which point in ETC ?

Best regards,

Tran Quang Hung

 

----------------------------------------------------

#1504

 

Hi Hung,

 

The perspector's first barycentric is:

 

(b+c)^2 (-a^14 b-3 a^13 b^2-2 a^12 b^3+2 a^11 b^4+5 a^10 b^5+7 a^9 b^6+4 a^8 b^7-4 a^7 b^8-7 a^6 b^9-5 a^5 b^10-2 a^4 b^11+2 a^3 b^12+3 a^2 b^13+a b^14-a^14 c-2 a^13 b c-8 a^12 b^2 c-16 a^11 b^3 c-8 a^10 b^4 c+4 a^9 b^5 c+9 a^8 b^6 c+16 a^7 b^7 c+11 a^6 b^8 c-2 a^5 b^9 c-2 a^4 b^10 c-2 a^2 b^12 c+b^14 c-3 a^13 c^2-8 a^12 b c^2-8 a^11 b^2 c^2-14 a^10 b^3 c^2-8 a^9 b^4 c^2+25 a^8 b^5 c^2+20 a^7 b^6 c^2-8 a^6 b^7 c^2+a^5 b^8 c^2+6 a^4 b^9 c^2-4 a^3 b^10 c^2-2 a^2 b^11 c^2+2 a b^12 c^2+b^13 c^2-2 a^12 c^3-16 a^11 b c^3-14 a^10 b^2 c^3+2 a^9 b^3 c^3-2 a^8 b^4 c^3+4 a^7 b^5 c^3+36 a^6 b^6 c^3+16 a^5 b^7 c^3-18 a^4 b^8 c^3-4 a^3 b^9 c^3+2 a^2 b^10 c^3-2 a b^11 c^3-2 b^12 c^3+2 a^11 c^4-8 a^10 b c^4-8 a^9 b^2 c^4-2 a^8 b^3 c^4+28 a^7 b^4 c^4-16 a^5 b^6 c^4+20 a^4 b^7 c^4+2 a^3 b^8 c^4-8 a^2 b^9 c^4-8 a b^10 c^4-2 b^11 c^4+5 a^10 c^5+4 a^9 b c^5+25 a^8 b^2 c^5+4 a^7 b^3 c^5+28 a^5 b^5 c^5-4 a^4 b^6 c^5-12 a^3 b^7 c^5-a^2 b^8 c^5-b^10 c^5+7 a^9 c^6+9 a^8 b c^6+20 a^7 b^2 c^6+36 a^6 b^3 c^6-16 a^5 b^4 c^6-4 a^4 b^5 c^6+32 a^3 b^6 c^6+8 a^2 b^7 c^6+5 a b^8 c^6-b^9 c^6+4 a^8 c^7+16 a^7 b c^7-8 a^6 b^2 c^7+16 a^5 b^3 c^7+20 a^4 b^4 c^7-12 a^3 b^5 c^7+8 a^2 b^6 c^7+4 a b^7 c^7+4 b^8 c^7-4 a^7 c^8+11 a^6 b c^8+a^5 b^2 c^8-18 a^4 b^3 c^8+2 a^3 b^4 c^8-a^2 b^5 c^8+5 a b^6 c^8+4 b^7 c^8-7 a^6 c^9-2 a^5 b c^9+6 a^4 b^2 c^9-4 a^3 b^3 c^9-8 a^2 b^4 c^9-b^6 c^9-5 a^5 c^10-2 a^4 b c^10-4 a^3 b^2 c^10+2 a^2 b^3 c^10-8 a b^4 c^10-b^5 c^10-2 a^4 c^11-2 a^2 b^2 c^11-2 a b^3 c^11-2 b^4 c^11+2 a^3 c^12-2 a^2 b c^12+2 a b^2 c^12-2 b^3 c^12+3 a^2 c^13+b^2 c^13+a c^14+b c^14)::

 

Not in ETC.

Searches {-2.45280231323039235286358699070,-4.21158485458085001844526817380,7.68843967964668712490212014160}.

 

The point D = {-(a+b-c) (a-b+c) (b+c)^2 (a b+b^2+a c+c^2)^2,(a+c)^2 (a-b+c) (a+b+c) (-a b-b^2+a c-c^2)^2,(a+b)^2 (a+b-c) (a+b+c) (a b-b^2-a c-c^2)^2}.

 

Best regards,

Peter Moses.

 

----------------------------------------------------

#1505

 

Hi Hung,

 

Forgot to mention that:

tangent at D,E,F to (Ia),(Ib),(Ic), reps bound a triangle which is perspective with the intangents triangle at

 

(a-b-c) (a+b-c) (a-b+c) X[8] + (a+b+c) (a^2+b^2+c^2) X[1854] =

 

(-a+b+c) (2 a^8 b+3 a^7 b^2+a^6 b^3+a^5 b^4-a^4 b^5-3 a^3 b^6-a^2 b^7-a b^8-b^9+2 a^8 c-2 a^7 b c-a^6 b^2 c+2 a^5 b^3 c-3 a^4 b^4 c+2 a^3 b^5 c+a^2 b^6 c-2 a b^7 c+b^8 c+3 a^7 c^2-a^6 b c^2-6 a^5 b^2 c^2+4 a^4 b^3 c^2+3 a^3 b^4 c^2-5 a^2 b^5 c^2+2 b^7 c^2+a^6 c^3+2 a^5 b c^3+4 a^4 b^2 c^3-4 a^3 b^3 c^3+5 a^2 b^4 c^3+2 a b^5 c^3-2 b^6 c^3+a^5 c^4-3 a^4 b c^4+3 a^3 b^2 c^4+5 a^2 b^3 c^4+2 a b^4 c^4-a^4 c^5+2 a^3 b c^5-5 a^2 b^2 c^5+2 a b^3 c^5-3 a^3 c^6+a^2 b c^6-2 b^3 c^6-a^2 c^7-2 a b c^7+2 b^2 c^7-a c^8+b c^8-c^9):: on lines

{{8,1854},{33,429},{221,388}}.

 

Best regards,

Peter Moses.

 

----------------------------------------------------

#1506

 

 

Dear Mr Peter,

Thank you so much for help me to find this point.

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. Please help me to check this.

Thank you very much and best regards,

Tran Quang Hung.

 

----------------------------------------------------

#1508

 

 

Hi Hung,

 

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)::

 

Center of (K) = {{72,171}, {191,1045}..} = a (a^6+2 a^5 b+2 a^4 b^2+2 a^3 b^3-2 a b^5-b^6+2 a^5 c+6 a^4 b c+8 a^3 b^2 c-6 a b^4 c-2 b^5 c+2 a^4 c^2+8 a^3 b c^2+a^2 b^2 c^2-8 a b^3 c^2-4 b^4 c^2+2 a^3 c^3-8 a b^2 c^3-6 b^3 c^3-6 a b c^4-4 b^2 c^4-2 a c^5-2 b c^5-c^6)::

 

Best regards,

Peter Moses