HYACINTHOS 28887

[Kadir Altintas]

 

Let ABC be a triangle and DEF the orthic triangle.

Let Xa, Xb, Xc be same centers of the triangles AFE, BFD, CDE, resp

 

The four circles (DXbXc), (EXaXc), (FXaXb) and NPC of ABC concur at a point X

 

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[Ercole Suppa]

 

Some points new points on NPC:

 

 

*** Xa, Xb, Xc = X(19) = Clawson points of their triangles then 

 

X = COMPLEMENT OF X(13395) 

 

= (-a+b+c) (b-c)^2 (a^2-b^2-c^2) (a^4-b^4+2 a^2 b c+2 a b^2 c+2 a b c^2+2 b^2 c^2-c^4) (a^4-2 a^2 b^2+b^4-4 a^2 b c-2 a^2 c^2-2 b^2 c^2+c^4) :: (barys)

 

= 3*X[2]-X[13395]

 

= lies on the nine point circle and the lines: {2,13395}, {119,5778}, {120,26066}, {122,26933}, {125,6506}, {127,26932}, {1146,5521}, {3270,15607}, {5798,25640}

 

= complement of X(13395)

 

= complementary conjugate of isogonal conjugate of X(13395)

 

= barycentric pproduct of X(377) and X(26956)

 

= (6-9-13) search numbers [3.58338282576268131, 3.01388104450312465, -0.0997375454082524181]

 

 

*** Xa, Xb, Xc = X(111) = Parry points of their triangles then 

 

X= COMPLEMENT OF X(6082)

 

= (b-c)^2 (b+c)^2 (-5 a^2+b^2+c^2) (-2 a^2+b^2+c^2) (a^2+b^2-3 b c+c^2) (a^2+b^2+3 b c+c^2) :: (barys)

 

= (972 R^4+81 R^2 SB+81 R^2 SC-243 R^2 SW+9 SW^2)S^4 + (-81 R^2 SB SC SW-54 R^2 SB SW^2-54 R^2 SC SW^2+9 SB SC SW^2-36 R^2 SW^3+7 SW^4)S^2 + 3 SB SC SW^4+2 SB SW^5+2 SC SW^5 :: (barys)

 

=  3*X[2]-X[6082], X[4]+X[6093], 2*X[5]-X[6092], X[14360]+X[14515]

= lies on the nine point circle and the lines: {2,6082}, {4,6093}, {5,6092}, {114,5913}, {126,524}, {1499,2686}, {1648,5099}, {6792,16188}, {8176,9169}, {8288,20383}, {14360,14515}

= midpoint of X(4) and X(6093)

= complement of X(6082)

 

= complementary conjugate of X(6088)

 

= perspector of circumconic centerd at X(9125)

 

= barycentric quotient of X(9125) and X(6082)

 

= reflection of X(i) in X(j) for these {i,j}: {6076,5512}, {6077,126}, {6092,5}, {6791,14858}

 

= (6-9-13) search numbers [-3.15456462695907193, -2.16531166197001418, 6.59567930648319135]

 

 

*** Xa, Xb, Xc = X(351) = center of the Parry circle of their triangles then 

 

X= COMPLEMENT OF X(2770) 

 

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

 

= (324 R^4-99 R^2 SW+7 SW^2)S^4 + (-27 R^2 SB SC SW-9 R^2 SB SW^2-9 R^2 SC SW^2+3 SB SC SW^2-6 R^2 SW^3+2 SB SW^3+2 SC SW^3+SW^4)S^2 + SB SC SW^4 :: (barys)

 

= X[4]+X[2696], 2*X[6698]-X[16339]

= lies on the nine point circle and the lines: {2,691}, {4,2696}, {30,5512}, {113,1499}, {115,858}, {125,524}, {126,523}, {127,5159}, {148,15398}, {373,2679}, {468,5139}, {1560,2489}, {2072,14672}, {2453,11336}, {3815,9193}, {5094,16221}, {5476,9169}, {5912,23991}, {6698,16339}, {7472,10418}, {8371,16188}, {9127,12494}, {11594,13994}, {14568,20389}, {16051,16177}

= midpoint of X(858) and X(5913)

 

= reflection of X(16339) in X(6698)

= complement of X(2770)

 

= complementary conjugate of X(2854)

 

= perspector of circumconic centerd at X(10418)

 

= barycentric quotient of X(10418) and X(2770) 

 

= (6-9-13) search numbers [-0.232165211958849069, 2.48024480331653639, 2.03072510051547655]

 

 

Best regards

Ercole Suppa