[Kadir Altintas]:
Let ABC be a triangle, P a point and DEF the circumcevian triangle of P.
Let Na,Nb,Nc be the NPC centers of the triangles BDC, CEA, AFB respectively
Let N'a,N'b,N'c be the NPC centers of the triangles AFE, BFD, CDE respectively
The triangles NaNbNc and Na'Nb'Nc' are perspective at a point Q with first barycentrics
Q = -a^10 x y^2 z^2 (2 x+y+z)-(b^2-c^2)^2 x^2 (y+z) (c^2 y+b^2 z) (b^4 (2 x+y) z+c^4 y (2 x+z)-b^2 c^2 (3 x^2+2 y z+2 x (y+z)))+a^8 y z (b^2 z (-2 x^3+7 x^2 y+y (y-z) z+4 x y (y+z))+c^2 y (-2 x^3+7 x^2 z+y z (-y+z)+4 x z (y+z)))+a^6 (-b^4 z^2 (8 x^2 y^2+2 y^3 z-x^3 (8 y+z)+x y (6 y^2+5 y z+3 z^2))-c^4 y^2 (8 x^2 z^2+2 y z^3-x^3 (y+8 z)+x z (3 y^2+5 y z+6 z^2))-b^2 c^2 y z (2 x^4-4 y^2 z^2-7 x^3 (y+z)+x (y^3+y^2 z+y z^2+z^3)))-a^4 (b^6 z^2 (-y^2 z (y+z)+4 x^3 (3 y+z)-2 x y (2 y^2+y z+z^2)+x^2 (-2 y^2+y z+2 z^2))+c^6 y^2 (-y z^2 (y+z)+4 x^3 (y+3 z)+x^2 (2 y^2+y z-2 z^2)-2 x z (y^2+y z+2 z^2))+b^4 c^2 z (-x^4 (7 y+z)+4 x^3 (2 y^2+y z+z^2)+y^2 z (-y^2+y z+2 z^2)+x^2 (y^3+6 y^2 z+6 y z^2+2 z^3)+x (-2 y^4+6 y^3 z+4 y z^3))+b^2 c^4 y (-x^4 (y+7 z)+y z^2 (2 y^2+y z-z^2)+4 x^3 (y^2+y z+2 z^2)+x^2 (2 y^3+6 y^2 z+6 y z^2+z^3)+x (4 y^3 z+6 y z^3-2 z^4)))+a^2 x (b^8 z^2 (-y^3+y z^2+x^2 (8 y+5 z)+2 x (y^2+y z+z^2))+c^8 y^2 (x^2 (5 y+8 z)+z (y^2-z^2)+2 x (y^2+y z+z^2))-b^2 c^6 y (4 x^3 (y+2 z)+x^2 (6 y^2+9 y z-3 z^2)+2 x (2 y^3+y^2 z+2 y z^2-z^3)+z (3 y^3-y^2 z-3 y z^2+z^3))-b^6 c^2 z (4 x^3 (2 y+z)+x^2 (-3 y^2+9 y z+6 z^2)+y (y^3-3 y^2 z-y z^2+3 z^3)+x (-2 y^3+4 y^2 z+2 y z^2+4 z^3))+b^4 c^4 (3 y (y-z)^2 z (y+z)-4 x^3 (y^2+z^2)+x^2 (y^3-2 y^2 z-2 y z^2+z^3)+2 x (y^4-y^3 z+2 y^2 z^2-y z^3+z^4))) : :
P=X(1) ---> Q=X(1125)
P=X(2) ---> Q=?
P=X(3) ---> Q=X(3)
P=X(4) ---> Q=X(389)
P=X(5) ---> Q=?
P=X(6) ---> Q=?
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[Ercole Suppa]
Let Q = Q(P) the perspector of NaNbNc and Na'Nb'Nc'
*** Some points Q = Q(P):
--- Q(X(2)) = COMPLEMENT OF X(12506)
= -4 a^10+13 a^8 (b^2+c^2)-3 a^6 (5 b^4-4 b^2 c^2+5 c^4)-6 (b^2-c^2)^2 (b^6-2 b^4 c^2-2 b^2 c^4+c^6)-a^4 (7 b^6+33 b^4 c^2+33 b^2 c^4+7 c^6)+a^2 (19 b^8-32 b^6 c^2-6 b^4 c^4-32 b^2 c^6+19 c^8) : : (barys)
= (54 R^2-21 SW)S^4 +(54 R^2 SB SC-9 SB SC SW+3 SB SW^2+3 SC SW^2-4 SW^3)S^2 -2 SB SC SW^3 : : (barys)
= 3*X[2]+X[12505], X[3]-3*X[10163], X[4]+3*X[9829], 5*X[1656]-3*X[10162], 7*X[3090]-3*X[6032], 5*X[3091]+3*X[6031], 4*X[3628]-3*X[10173]
= lies on these lines: {2,12505}, {3,10163}, {4,9829}, {5,3849}, {1656,10162}, {3090,6032}, {3091,6031}, {3628,10173}, {3934,8704}, {7550,14682}
= complement of X(12506)
= midpoint of X(i) and X(j) for these {i,j}: {3,14866}, {12505,12506}
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {2,12505,12506}, {10163,14866,3}
= (6-8-13) search numbers [3.20626132412208190, 1.11830763447042296, 1.38663858537158695]
--- Q(X(5)) = (name pending)
= a^20 (b^2+c^2)-(b^2-c^2)^10 (b^2+c^2)-2 a^18 (4 b^4+5 b^2 c^2+4 c^4)+3 a^16 (9 b^6+11 b^4 c^2+11 b^2 c^4+9 c^6)+2 a^2 (b^2-c^2)^6 (4 b^8-2 b^6 c^2-5 b^4 c^4-2 b^2 c^6+4 c^8)-4 a^14 (12 b^8+10 b^6 c^2+9 b^4 c^4+10 b^2 c^6+12 c^8)+2 a^10 b^2 c^2 (25 b^8+13 b^6 c^2+11 b^4 c^4+13 b^2 c^6+25 c^8)-a^4 (b^2-c^2)^4 (27 b^10-7 b^8 c^2-29 b^6 c^4-29 b^4 c^6-7 b^2 c^8+27 c^10)+a^12 (42 b^10-8 b^8 c^2-17 b^6 c^4-17 b^4 c^6-8 b^2 c^8+42 c^10)+12 a^6 (b^2-c^2)^2 (4 b^12-b^10 c^2-3 b^8 c^4-4 b^6 c^6-3 b^4 c^8-b^2 c^10+4 c^12)+a^8 (-42 b^14+10 b^12 c^2+28 b^10 c^4+22 b^8 c^6+22 b^6 c^8+28 b^4 c^10+10 b^2 c^12-42 c^14) : : (barys)
= (13 R^2-4 SW)S^4 + (-R^6-5 R^4 SB-5 R^4 SC+17 R^2 SB SC+2 R^4 SW+2 R^2 SB SW+2 R^2 SC SW-4 SB SC SW-R^2 SW^2)S^2 + R^6 SB SC -R^2 SB SC SW^2 : : (barys)
= lies on this line: {5,252}
= (6-8-13) search numbers [-0.564361202838121525, -2.25177783291942595, 5.46006199831542239]
--- Q(X(6)) = MIDPOINT OF X(3) AND X(14867)
= -4 a^10+15 a^8 (b^2+c^2)+3 a^2 (b^2-c^2)^2 (2 b^4+b^2 c^2+2 c^4)-4 a^6 (5 b^4+2 b^2 c^2+5 c^4)-(b^2-c^2)^2 (2 b^6-3 b^4 c^2-3 b^2 c^4+2 c^6)+a^4 (5 b^6-21 b^4 c^2-21 b^2 c^4+5 c^6) : : (barys)
= (54 R^2-9 SB-9 SC-18 SW)S^4 + (54 R^2 SB SC-3 SB SW^2-3 SC SW^2)S^2 -2 SB SC SW^3
= X[3]-3*X[10166], X[4]+3*X[353], 2*X[140]-3*X[10160]
= lies on these lines: {3,10166}, {4,353}, {5,9830}, {6,22100}, {39,11615}, {83,6233}, {140,10160}, {1499,12506}, {1506,12494}, {8705,15074}
= midpoint of X(3) and X(14867)
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {10166,14867,3}
= (6-8-13) search numbers [1.04380990313095110, 1.77913705910783955, 1.92719579338774353]
Best regards
Ercole Suppa
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