HYACINTHOS 29480

[Kadir Altintas]

Let ABC be a triangle.

 

Denote:

Fe = the Feuerbach point X(11)

Na =  Nagel point X(8)
DEF = the cevian triangle of Na
Ha = the orthocenter od FeFE. 

Define Hb, Hc cyclically

Prove:  AHa, BHb, CHc concurr at a point X

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[Ercole Suppa]
 
X = ISOGONAL CONJUGATE OF X(13528) =

= a (a^6-2 a^5 b-a^4 b^2+4 a^3 b^3-a^2 b^4-2 a b^5+b^6+a^5 c+5 a^4 b c-6 a^3 b^2 c-6 a^2 b^3 c+5 a b^4 c+b^5 c-4 a^4 c^2+4 a^3 b c^2+8 a^2 b^2 c^2+4 a b^3 c^2-4 b^4 c^2-2 a^3 c^3-6 a^2 b c^3-6 a b^2 c^3-2 b^3 c^3+5 a^2 c^4-2 a b c^4+5 b^2 c^4+a c^5+b c^5-2 c^6) (a^6+a^5 b-4 a^4 b^2-2 a^3 b^3+5 a^2 b^4+a b^5-2 b^6-2 a^5 c+5 a^4 b c+4 a^3 b^2 c-6 a^2 b^3 c-2 a b^4 c+b^5 c-a^4 c^2-6 a^3 b c^2+8 a^2 b^2 c^2-6 a b^3 c^2+5 b^4 c^2+4 a^3 c^3-6 a^2 b c^3+4 a b^2 c^3-2 b^3 c^3-a^2 c^4+5 a b c^4-4 b^2 c^4-2 a c^5+b c^5+c^6) : : (barys)

 

= 2*X[11]-X[10309]  

= lies on the Feuerbach circumhyperbola and these lines: {4,24465}, {8,2829}, {11,10309}, {515,12641}, {1000,6938}, {1320,6001}, {2800,3680}, {6850,33898}

= isogonal conjugate of X(13528) 

= antigonal image of  X(10309)  

= (6-9-13) search numbers [369.7499131674740827415,436.6661708397088483733,-469.3204906382636310360]


Best regards,
Ercole Suppa

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[Kadir Altintas]:

Let ABC be a triangle.

 

Denote:

Fe = the Feuerbach point X(11)

Na =  Nagel point X(8)
DEF = the cevian triangle of Na

Ka = the symmedian point of FeFE. 

Define Kb, Kc cyclically

Prove:  AKa, BKb, CKc concur at a point X

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[Ercole Suppa]
 
X = pending name

= (b-c)^2 (-a+b+c) (a^4-2 a^2 b^2+b^4+2 a^2 b c+2 a b^2 c-2 a^2 c^2-2 b^2 c^2+c^4) (a^4-2 a^2 b^2+b^4+2 a^2 b c-2 a^2 c^2+2 a b c^2-2 b^2 c^2+c^4) : : (barys)

= (6-9-13) search numbers [2.2559647749156663191, 3.5631205896053029706, 0.1326741776042464378]


Best regards,
Ercole Suppa