HYACINTHOS 29115

[Antreas P. Hatzipolakis]:

Let ABC be a triangle and P a point.

Denote:

Na, Nb, Nc = the NPC centers of PBC, PCA, PAB, resp.

The perpenduculars from Na, Nb, Nc to AP, BP, CP, resp. concur at a point Q.

Which is the locus of P such that Q lies on the Euler line of ABC?
G lies on the locus.

APH

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[Ercole Suppa]:

Hi Antreas,

*** The locus of points P such that Q = Q(P) lies on the Euler line of ABC is the curve

q4: ∑ (b-c) (b+c) (a^2-b^2-c^2) x^2 y z-a^2 (b-c) (b+c) y^2 z^2 = 0

circumquartic through ETC centers X(i) for these i: {2,4,54,251,847,943,961,1113,1114,1166,1487,5627,10415,10419,10422,18018,19307}


*** Pairs {P = X(i),Q = X(j)} for these {i,j}: {2, 3845}, {4, 5}

*** Some points:


Q(X(54)) = COMPLEMENT OF X(13564)

= a^8 b^2-2 a^6 b^4+2 a^2 b^8-b^10+a^8 c^2-4 a^6 b^2 c^2-a^4 b^4 c^2+a^2 b^6 c^2+3 b^8 c^2-2 a^6 c^4-a^4 b^2 c^4-6 a^2 b^4 c^4-2 b^6 c^4+a^2 b^2 c^6-2 b^4 c^6+2 a^2 c^8+3 b^2 c^8-c^10 : : (barys)

= S^2 (7 R^2 - 2 SW) -R^2 SB SC - 2 SB SC SW : : (barys)

As a point on the Euler line, X() has Shinagawa coefficients [e+8f,9e+8f]

= lies on these lines: {2,3}, {51,13561}, {125,10095}, {389,10264}, {511,6153}, {542,1493}, {1173,9140}, {1209,13391}, {1503,10274}, {3448,14627}, {3574,5663}, {5944,13419}, {5946,20299}, {6000,32351}, {6689,29012}, {6748,9380}, {7703,9781}, {10263,21243}, {10280,23301}, {10540,15806}, {11550,32046}, {13754,20424}, {13851,15807}, {14676,32134}, {15026,19130}, {16105,20304}, {17702,22804}, {22330,25328}, {23315,32767}, {23319,25147}, {31664,31665}

= complement of X(13564)

= reflection of X(5) in X(5576)

= (6-9-13)  search numbers:  [-61.75518910567356589521, -62.46179728041226395262, 75.38584218634989994502]


Q(X(251)) = X(2)X(3) ∩ X(31664)X(31665)

= a^8 b^2-b^10+a^8 c^2+8 a^6 b^2 c^2+3 a^4 b^4 c^2-a^2 b^6 c^2+b^8 c^2+3 a^4 b^2 c^4+10 a^2 b^4 c^4-a^2 b^2 c^6+b^2 c^8-c^10 : : (barys)

= (R^2-SW)S^4 + (9 R^2 SB SC-SB SC SW+SW^3)S^2 + SB SC SW^3 : : (barys)

As a point on the Euler line, X() has Shinagawa coefficients [R^2 S^2-S^2 SW+SW^3,9 R^2 S^2-S^2 SW+SW^3]

= lies on these lines: {2,3}, {31664,31665}


= (6-9-13)  search numbers:  [0.036657098581306240159, -0.83295952666795677106, 4.20041010871697706455]


Q(X(943)) = X(2)X(3) ∩ X(5806)X(10265)

= a^8 b^2-2 a^7 b^3-2 a^6 b^4+6 a^5 b^5-6 a^3 b^7+2 a^2 b^8+2 a b^9-b^10-4 a^8 b c+2 a^7 b^2 c+5 a^6 b^3 c-a^5 b^4 c+4 a^4 b^5 c-4 a^3 b^6 c-7 a^2 b^7 c+3 a b^8 c+2 b^9 c+a^8 c^2+2 a^7 b c^2+8 a^6 b^2 c^2+5 a^5 b^3 c^2-2 a^4 b^4 c^2+2 a^3 b^5 c^2-10 a^2 b^6 c^2-9 a b^7 c^2+3 b^8 c^2-2 a^7 c^3+5 a^6 b c^3+5 a^5 b^2 c^3-4 a^4 b^3 c^3+8 a^3 b^4 c^3+7 a^2 b^5 c^3-11 a b^6 c^3-8 b^7 c^3-2 a^6 c^4-a^5 b c^4-2 a^4 b^2 c^4+8 a^3 b^3 c^4+16 a^2 b^4 c^4+15 a b^5 c^4-2 b^6 c^4+6 a^5 c^5+4 a^4 b c^5+2 a^3 b^2 c^5+7 a^2 b^3 c^5+15 a b^4 c^5+12 b^5 c^5-4 a^3 b c^6-10 a^2 b^2 c^6-11 a b^3 c^6-2 b^4 c^6-6 a^3 c^7-7 a^2 b c^7-9 a b^2 c^7-8 b^3 c^7+2 a^2 c^8+3 a b c^8+3 b^2 c^8+2 a c^9+2 b c^9-c^10 : : (barys)

As a point on the Euler line, X() has Shinagawa coefficients [r(2r+5 R), 2 r^2 + 13 r R + 16 R^2]

= lies on these lines: {2,3}, {5806,10265}, {20116,20330}, {31664,31665}

= (6-9-13)  search numbers:  [-4.31745254311159175714, -5.17558291718623623210, 9.21643075216481576709]


Q(X(961)) = Q(X(251))


Best regards,
Ercole Suppa