HYACINTHOS 28882

[Kadir Altintas]:

 

Let ABC be a triangle and DEF the circumcevian triangle of the incenter I.

Let Xa, Xb, Xc be same points of triangles BDC, ACE, ABF, resp. lying on their Euler lines.

 

(1) Conjecture: 

 

If P is same point on Euler lines of triangles BDC, ACE, ABF such P=O + t H (t=number), then the NPC center X of triangle XaXbXc lies on the Euler line of ABC.

 

(2) For X(2) and X(20) of BDC, ACE, ABF they are new centers on Euler line of ABC.

 

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[Ercole Suppa]

 

 

(2) some points:

 

 

*** for X(2) of BDC, ACE, ABF the NPC center of XaXbXc is

 

X= EULER LINE INTERCEPT OF X(2)X(28443)

 

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

 

= 3 R S^3 + (12 a R^2-12 b R^2-5 a SB+5 b SB-5 a SC+5 c SC+2 a SW+3 b SW) S^2 +3 R S SB SC+3 b SB SC^2-3 c SB SC^2-3 b SB SC SW :: (barys)

 

= 2*X[1125]+X[22937], X[1483]+2*X[21677], 7*X[3624]-X[16159], X[5426]+X[26446], X[10264]+2*X[16164]

As a point on the Euler line, X has Shinagawa coefficients {10 r + 13 R, -3 (2 r + R)}

= lies on these lines: {2,3}, {495,5427}, {551,6583}, {1125,22937}, {1483,21677}, {1749,5444}, {2771,10165}, {3337,5298}, {3624,16159}, {3656,24468}, {4860,16137}, {5251,11698}, {5426,26446}, {5441,12019}, {5536,5901}, {10264,16164}, {10543,18395}, {10573,15174}, {15254,22936}

 

= midpoint of X(i) and X(j) for these {i,j}: {2,28443}, {5054,15672}, {5426,26446}, {15670,28465}

 

= reflection of X(549) in X(28465)

 

= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {3,10021,16160}, {3,15674,10021}, {21,140,5499}, {549,632,17564}, {631,13743,11277}, {631,15676,13743}, {3523,16117,11276}, {5428,6675,5}

 

= (6-8-13) search numbers [3.50413383204873372, 2.62536991562393952, 0.205808156299159255]

 

 

*** for X(4) of BDC, ACE, ABF the NPC center of XaXbXc is X=X(16160)

 

*** for X(5) of BDC, ACE, ABF the NPC center of XaXbXc is X=X(10021)

 

 

*** for X(20) of BDC, ACE, ABF the NPC center of XaXbXc is

 

X= EULER LINE INTERCEPT OF X(79)X(5719)

 

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

 

= 5 R S^3 + (20 a R^2-20 b R^2-3 a SB+3 b SB-3 a SC+3 c SC-2 a SW+5 b SW)S^2 -11 R S SB SC+5 b SB SC^2-5 c SB SC^2-5 b SB SC SW :: (barys)

 

= 2*X[12512]-X[22937]

 

As a point on the Euler line, X has Shinagawa coefficients {6 r + 11 R, -10 r - 21 R}

= lies on these on lines: {2,3}, {79,5719}, {2771,4067}, {3337,5441}, {3648,3940}, {4297,6583}, {5761,16150}, {10123,24929}, {10386,16137}, {10543,15935}, {11263,28146}, {11374,16118}, {12512,22937}, {16113,24466}, {18481,24468}

 

= reflection of X(i) in X(j) for these {i,j}: {4,11277}, {21,548}, {3627,442}, {16160,3},{22937,12512}, {28460,15690}

 

= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {3,382,6884}, {3,6894,140}

 

= (6-8-13) search numbers [16.6170500806391810, 15.7036939600537939, -14.9005313738094079]

 

 

Best regards

Ercole Suppa