HYACINTHOS 28984

[J.L. Ayme] 

 

1. Let ABC be a triangle

2. H=X(4), O=X(3), Na=X(8) orthocenter, circumcenter, Nagel point of ABC, resp.

3. PQR orthic triangle of ABC

4. L Euler line of PQR

5. X intersection point of line parallel to L through H and line ONa.

 

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[Ercole Suppa]

 

X= X(3)X(8) ∩ X(4)X(93)

 

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

 

= lies on these lines: {3,8}, {4,93}, {498,2594}, {6928,10449}, {10441,10526}

 

= (6-9-13) search numbers [-1.20400705414531219, -8.20688496760562098, 9.87804963831687618]

 

Best regards

Ercole Suppa