[Antreas P. Hatzipolakis]:
Let ABC be a triangle and P a point.
Denote:
Na, Nb, Nc = the NPC centers of PBC, PCA, PAB, resp.
Which is the locus of P such that:
1. the reflections of PNa, PNb, PNc in NbNc, NcNa, NaNb, resp. are concurrent?
I lies on the locus (trivial case since I of ABC = H of NaNbNc).
I think that O, N, N* = isogonal conjugate of N = X(54) lie on the locus as well
(points of concurrence?)
So there is evidence that the locus is K005 = the Napoleon - Feuerbach cubic ( + other things)
So there is evidence that the locus is K005 = the Napoleon - Feuerbach cubic ( + other things)
[Peter Moses]:
Hi Antreas,
I make the locus:
Inf + Napoleon - Feuerbach cubic K005
+
(c^4 x^2 y^2+a^4 x^2 y z-2 a^2 b^2 x^2 y z+b^4 x^2 y z-2 a^2 c^2 x^2 y z+c^4 x^2 y z+a^4 x y^2 z-2 a^2 b^2 x y^2 z+b^4 x y^2 z-2 b^2 c^2 x y^2 z+c^4 x y^2 z+b^4 x^2 z^2+a^4 x y z^2+b^4 x y z^2-2 a^2 c^2 x y z^2-2 b^2 c^2 x y z^2+c^4 x y z^2+a^4 y^2 z^2) = 0
through X(4)
+
(a^4 c^4 x^5 y^3-2 a^2 b^2 c^4 x^5 y^3+b^4 c^4 x^5 y^3-2 a^2 c^6 x^5 y^3+b^2 c^6 x^5 y^3+c^8 x^5 y^3+2 a^4 c^4 x^4 y^4-4 a^2 b^2 c^4 x^4 y^4+2 b^4 c^4 x^4 y^4-a^2 c^6 x^4 y^4-b^2 c^6 x^4 y^4-c^8 x^4 y^4+a^4 c^4 x^3 y^5-2 a^2 b^2 c^4 x^3 y^5+b^4 c^4 x^3 y^5+a^2 c^6 x^3 y^5-2 b^2 c^6 x^3 y^5+c^8 x^3 y^5-3 a^2 b^2 c^4 x^5 y^2 z+6 b^4 c^4 x^5 y^2 z+3 b^2 c^6 x^5 y^2 z+2 a^4 c^4 x^4 y^3 z-a^2 b^2 c^4 x^4 y^3 z+8 b^4 c^4 x^4 y^3 z-4 a^2 c^6 x^4 y^3 z-10 b^2 c^6 x^4 y^3 z+2 c^8 x^4 y^3 z+8 a^4 c^4 x^3 y^4 z-a^2 b^2 c^4 x^3 y^4 z+2 b^4 c^4 x^3 y^4 z-10 a^2 c^6 x^3 y^4 z-4 b^2 c^6 x^3 y^4 z+2 c^8 x^3 y^4 z+6 a^4 c^4 x^2 y^5 z-3 a^2 b^2 c^4 x^2 y^5 z+3 a^2 c^6 x^2 y^5 z-3 a^2 b^4 c^2 x^5 y z^2+3 b^6 c^2 x^5 y z^2+6 b^4 c^4 x^5 y z^2+a^8 x^4 y^2 z^2-4 a^6 b^2 x^4 y^2 z^2+6 a^4 b^4 x^4 y^2 z^2-4 a^2 b^6 x^4 y^2 z^2+b^8 x^4 y^2 z^2-4 a^6 c^2 x^4 y^2 z^2+a^4 b^2 c^2 x^4 y^2 z^2+4 a^2 b^4 c^2 x^4 y^2 z^2-b^6 c^2 x^4 y^2 z^2+6 a^4 c^4 x^4 y^2 z^2+4 a^2 b^2 c^4 x^4 y^2 z^2-4 a^2 c^6 x^4 y^2 z^2-b^2 c^6 x^4 y^2 z^2+c^8 x^4 y^2 z^2+2 a^8 x^3 y^3 z^2-8 a^6 b^2 x^3 y^3 z^2+12 a^4 b^4 x^3 y^3 z^2-8 a^2 b^6 x^3 y^3 z^2+2 b^8 x^3 y^3 z^2-8 a^6 c^2 x^3 y^3 z^2+8 a^4 b^2 c^2 x^3 y^3 z^2+8 a^2 b^4 c^2 x^3 y^3 z^2-8 b^6 c^2 x^3 y^3 z^2+13 a^4 c^4 x^3 y^3 z^2-18 a^2 b^2 c^4 x^3 y^3 z^2+13 b^4 c^4 x^3 y^3 z^2-10 a^2 c^6 x^3 y^3 z^2-10 b^2 c^6 x^3 y^3 z^2+3 c^8 x^3 y^3 z^2+a^8 x^2 y^4 z^2-4 a^6 b^2 x^2 y^4 z^2+6 a^4 b^4 x^2 y^4 z^2-4 a^2 b^6 x^2 y^4 z^2+b^8 x^2 y^4 z^2-a^6 c^2 x^2 y^4 z^2+4 a^4 b^2 c^2 x^2 y^4 z^2+a^2 b^4 c^2 x^2 y^4 z^2-4 b^6 c^2 x^2 y^4 z^2+4 a^2 b^2 c^4 x^2 y^4 z^2+6 b^4 c^4 x^2 y^4 z^2-a^2 c^6 x^2 y^4 z^2-4 b^2 c^6 x^2 y^4 z^2+c^8 x^2 y^4 z^2+3 a^6 c^2 x y^5 z^2-3 a^4 b^2 c^2 x y^5 z^2+6 a^4 c^4 x y^5 z^2+a^4 b^4 x^5 z^3-2 a^2 b^6 x^5 z^3+b^8 x^5 z^3-2 a^2 b^4 c^2 x^5 z^3+b^6 c^2 x^5 z^3+b^4 c^4 x^5 z^3+2 a^4 b^4 x^4 y z^3-4 a^2 b^6 x^4 y z^3+2 b^8 x^4 y z^3-a^2 b^4 c^2 x^4 y z^3-10 b^6 c^2 x^4 y z^3+8 b^4 c^4 x^4 y z^3+2 a^8 x^3 y^2 z^3-8 a^6 b^2 x^3 y^2 z^3+13 a^4 b^4 x^3 y^2 z^3-10 a^2 b^6 x^3 y^2 z^3+3 b^8 x^3 y^2 z^3-8 a^6 c^2 x^3 y^2 z^3+8 a^4 b^2 c^2 x^3 y^2 z^3-18 a^2 b^4 c^2 x^3 y^2 z^3-10 b^6 c^2 x^3 y^2 z^3+12 a^4 c^4 x^3 y^2 z^3+8 a^2 b^2 c^4 x^3 y^2 z^3+13 b^4 c^4 x^3 y^2 z^3-8 a^2 c^6 x^3 y^2 z^3-8 b^2 c^6 x^3 y^2 z^3+2 c^8 x^3 y^2 z^3+3 a^8 x^2 y^3 z^3-10 a^6 b^2 x^2 y^3 z^3+13 a^4 b^4 x^2 y^3 z^3-8 a^2 b^6 x^2 y^3 z^3+2 b^8 x^2 y^3 z^3-10 a^6 c^2 x^2 y^3 z^3-18 a^4 b^2 c^2 x^2 y^3 z^3+8 a^2 b^4 c^2 x^2 y^3 z^3-8 b^6 c^2 x^2 y^3 z^3+13 a^4 c^4 x^2 y^3 z^3+8 a^2 b^2 c^4 x^2 y^3 z^3+12 b^4 c^4 x^2 y^3 z^3-8 a^2 c^6 x^2 y^3 z^3-8 b^2 c^6 x^2 y^3 z^3+2 c^8 x^2 y^3 z^3+2 a^8 x y^4 z^3-4 a^6 b^2 x y^4 z^3+2 a^4 b^4 x y^4 z^3-10 a^6 c^2 x y^4 z^3-a^4 b^2 c^2 x y^4 z^3+8 a^4 c^4 x y^4 z^3+a^8 y^5 z^3-2 a^6 b^2 y^5 z^3+a^4 b^4 y^5 z^3+a^6 c^2 y^5 z^3-2 a^4 b^2 c^2 y^5 z^3+a^4 c^4 y^5 z^3+2 a^4 b^4 x^4 z^4-a^2 b^6 x^4 z^4-b^8 x^4 z^4-4 a^2 b^4 c^2 x^4 z^4-b^6 c^2 x^4 z^4+2 b^4 c^4 x^4 z^4+8 a^4 b^4 x^3 y z^4-10 a^2 b^6 x^3 y z^4+2 b^8 x^3 y z^4-a^2 b^4 c^2 x^3 y z^4-4 b^6 c^2 x^3 y z^4+2 b^4 c^4 x^3 y z^4+a^8 x^2 y^2 z^4-a^6 b^2 x^2 y^2 z^4-a^2 b^6 x^2 y^2 z^4+b^8 x^2 y^2 z^4-4 a^6 c^2 x^2 y^2 z^4+4 a^4 b^2 c^2 x^2 y^2 z^4+4 a^2 b^4 c^2 x^2 y^2 z^4-4 b^6 c^2 x^2 y^2 z^4+6 a^4 c^4 x^2 y^2 z^4+a^2 b^2 c^4 x^2 y^2 z^4+6 b^4 c^4 x^2 y^2 z^4-4 a^2 c^6 x^2 y^2 z^4-4 b^2 c^6 x^2 y^2 z^4+c^8 x^2 y^2 z^4+2 a^8 x y^3 z^4-10 a^6 b^2 x y^3 z^4+8 a^4 b^4 x y^3 z^4-4 a^6 c^2 x y^3 z^4-a^4 b^2 c^2 x y^3 z^4+2 a^4 c^4 x y^3 z^4-a^8 y^4 z^4-a^6 b^2 y^4 z^4+2 a^4 b^4 y^4 z^4-a^6 c^2 y^4 z^4-4 a^4 b^2 c^2 y^4 z^4+2 a^4 c^4 y^4 z^4+a^4 b^4 x^3 z^5+a^2 b^6 x^3 z^5+b^8 x^3 z^5-2 a^2 b^4 c^2 x^3 z^5-2 b^6 c^2 x^3 z^5+b^4 c^4 x^3 z^5+6 a^4 b^4 x^2 y z^5+3 a^2 b^6 x^2 y z^5-3 a^2 b^4 c^2 x^2 y z^5+3 a^6 b^2 x y^2 z^5+6 a^4 b^4 x y^2 z^5-3 a^4 b^2 c^2 x y^2 z^5+a^8 y^3 z^5+a^6 b^2 y^3 z^5+a^4 b^4 y^3 z^5-2 a^6 c^2 y^3 z^5-2 a^4 b^2 c^2 y^3 z^5+a^4 c^4 y^3 z^5) = 0
through X{13,14,110}.
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O -> X(10264).
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N -> X(14051).
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X(13) ->
= ISOGONAL CONJUGATE OF X(9203)
(Sqrt[3] a^2+Sqrt[3] b^2-Sqrt[3] c^2-2 S) (Sqrt[3] a^2-Sqrt[3] b^2+Sqrt[3] c^2-2 S) (5 a^2-b^2-c^2-2 Sqrt[3] S) (Sqrt[3] (b^2-c^2) (-a^2+b^2+c^2)-2 (b^2-c^2) S) : :
= lies on these lines: {2,9162}, {13,9180}, {14,5466}, {15,9147}, {30,511}, {98,2378}, {99,9202}, {115,11625}, {351,9194}, {619,1649}, {623,9148}, {691,23896}, {842,11613}, {1637,11627}, {5460,8371}, {5464,9168}, {5479,23283}, {5608,6109}, {5652,22689}, {5996,6114}, {6671,11176}, {7684,19912}, {8594,9485}, {9123,13304}, {9191,9205}, {14174,22687}, {14176,14181}, {14184,14187}, {14817,16220}, {15342,23895}, {25152,25172}, {25153,25174}, {25155,25176}, {25207,25210}, {25212,25216}, {25215,25229}, {25221,25231}, {25225,25233}
= barycentric product X(21467)*X(23871)
= barycentric quotient X(i)/X(j) for these {i,j}: {6, 9203}, {13304, 396}, {21467, 23896}
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X(14) ->
= ISOGONAL CONJUGATE OF X(9202)
(Sqrt[3] a^2+Sqrt[3] b^2-Sqrt[3] c^2+2 S) (Sqrt[3] a^2-Sqrt[3] b^2+Sqrt[3] c^2+2 S) (5 a^2-b^2-c^2+2 Sqrt[3] S) (Sqrt[3] (b^2-c^2) (-a^2+b^2+c^2)+2 (b^2-c^2) S) : :
= lies on these lines: {2,9163}, {13,5466}, {14,9180}, {16,9147}, {30,511}, {98,2379}, {99,9203}, {115,11627}, {351,9195}, {618,1649}, {624,9148}, {691,23895}, {842,11612}, {1637,11625}, {5459,8371}, {5463,9168}, {5478,23284}, {5607,6108}, {5652,22687}, {5996,6115}, {6672,11176}, {7685,19912}, {8595,9485}, {9123,13305}, {9191,9204}, {14175,14177}, {14180,22689}, {14183,14185}, {14816,16220}, {15342,23896}, {25162,25171}, {25163,25179}, {25165,25181}, {25208,25209}, {25211,25213}, {25218,25230}, {25222,25232}, {25226,25234}
= isogonal conjugate of X(9202)
= barycentric product X(21466)*X(23870)
= barycentric quotient X(i)/X(j) for these {i,j}: {6, 9202}, {13305, 395}, {21466, 23895}
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X(54) ->
= X(54)X(5498)∩X(125)X(8254)
2 a^16-9 a^14 b^2+15 a^12 b^4-9 a^10 b^6-5 a^8 b^8+13 a^6 b^10-11 a^4 b^12+5 a^2 b^14-b^16-9 a^14 c^2+18 a^12 b^2 c^2-3 a^10 b^4 c^2-8 a^8 b^6 c^2-7 a^6 b^8 c^2+18 a^4 b^10 c^2-13 a^2 b^12 c^2+4 b^14 c^2+15 a^12 c^4-3 a^10 b^2 c^4-8 a^8 b^4 c^4-9 a^6 b^6 c^4+9 a^2 b^10 c^4-4 b^12 c^4-9 a^10 c^6-8 a^8 b^2 c^6-9 a^6 b^4 c^6-14 a^4 b^6 c^6-a^2 b^8 c^6-4 b^10 c^6-5 a^8 c^8-7 a^6 b^2 c^8-a^2 b^6 c^8+10 b^8 c^8+13 a^6 c^10+18 a^4 b^2 c^10+9 a^2 b^4 c^10-4 b^6 c^10-11 a^4 c^12-13 a^2 b^2 c^12-4 b^4 c^12+5 a^2 c^14+4 b^2 c^14-c^16 : :
=3 X[11245] - X[22051]
= lies on these lines: {54,5498}, {125,8254}, {3530,10610}, {5012,21230}, {10116,11802}, {11245,22051}
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X(110) -> X(30).
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Best regards,
Peter Moses.
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