HYACINTHOS 29517

[Kadir Altintas]

Let ABC be a triangle.

Denote:

Fe = The Feuerbach pointX(11)
Na = The Nagel point X(8)
DEF = the cevian triangle of Na
Ha = the orthocenter of FeEF. 

Define Hb, Hc cyclically

Prove:  DHa, EHb, FHc concurr at a point X

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[Ercole Suppa]
 
X = X(8)X(153) ∩ X(11)X(18239) =

= a (a^11 b-3 a^10 b^2-a^9 b^3+11 a^8 b^4-6 a^7 b^5-14 a^6 b^6+14 a^5 b^7+6 a^4 b^8-11 a^3 b^9+a^2 b^10+3 a b^11-b^12+a^11 c-2 a^10 b c+6 a^9 b^2 c-9 a^8 b^3 c-18 a^7 b^4 c+40 a^6 b^5 c+8 a^5 b^6 c-46 a^4 b^7 c+9 a^3 b^8 c+18 a^2 b^9 = c-6 a b^10 c-b^11 c-3 a^10 c^2+6 a^9 b c^2+8 a^8 b^2 c^2-14 a^6 b^4 c^2-36 a^5 b^5 c^2+24 a^4 b^6 c^2+48 a^3 b^7 c^2-23 a^2 b^8 c^2-18 a b^9 c^2+8 b^10 c^2-a^9 c^3-9 a^8 b c^3+24 a^6 b^3 c^3+6 a^5 b^4 c^3+22 a^4 b^5 c^3-32 a^3 b^6 c^3-40 a^2 b^7 c^3+27 a b^8 c^3+3 b^9 c^3+11 a^8 c^4-18 a^7 b c^4-14 a^6 b^2 c^4+6 a^5 b^3 c^4-12 a^4 b^4 c^4-14 a^3 b^5 c^4+22 a^2 b^6 c^4+42 a b^7 c^4-23 b^8 c^4-6 a^7 c^5+40 a^6 b c^5-36 a^5 b^2 c^5+22 a^4 b^3 c^5-14 a^3 b^4 c^5+44 a^2 b^5 c^5-48 a b^6 c^5-2 b^7 c^5-14 a^6 c^6+8 a^5 b c^6+24 a^4 b^2 c^6-32 a^3 b^3 c^6+22 a^2 b^4 c^6-48 a b^5 c^6+32 b^6 c^6+14 a^5 c^7-46 a^4 b c^7+48 a^3 b^2 c^7-40 a^2 b^3 c^7+42 a b^4 c^7-2 b^5 c^7+6 a^4 c^8+9 a^3 b c^8-23 a^2 b^2 c^8+27 a b^3 c^8-23 b^4 c^8-11 a^3 c^9+18 a^2 b c^9-18 a b^2 c^9+3 b^3 c^9+a^2 c^10-6 a b c^10+8 b^2 c^10+3 a c^11-b c^11-c^12) : : (barys)

= X[11]+X[18239], X[5083]-2*X[12608], 2*X[6667]-X[18238], X[14740]-2*X[32159]

= lies on these lines: {8,153}, {11,18239}, {515,15558}, {960,2829}, {971,6713}, {1490,10058}, {2801,24389}, {3036,6001}, {5083,12608}, {6260,10265}, {6261,11715}, {6667,18238}, {14740,32159}

= reflection of X(14740) in X(32159) 

= (6-9-13) search numbers [27.7024812436264501141, 30.8609300917119886234, -30.5103553863361284580]


Best regards,
Ercole Suppa