HYACINTHOS 29428

[Antreas P. Hatzipolakis]:

 

 

Let ABC be a triangle, P, Q two isogonal conjugate points and A'B'C' the pedal triangle of P.

 

Denote:

 

Na, Nb, Nc = the NPC centers of QBC, QCA, QAB, resp.

 

Which is the locus of P such that A'B'C', NaNbNc are:

1. Orthologic ?

2. Parallelogic?

3. Perspective?

 

[Ercole Suppa] 

 

[...]¨

Q(X(74)) = ISOGONAL CONJUGATE OF X(15469)

 

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

 

= S^4 + (-18 R^2 SB-18 R^2 SC-3 SB SC+12 R^2 SW+4 SB SW+4 SC SW-3 SW^2)S^2 -324 R^4 SB SC+144 R^2 SB SC SW-15 SB SC SW^2 : : (barys)

 

= 2*X[113]-X[14611], 2*X[140]-3*X[21315], X[477]-2*X[3154], 2*X[548]-X[21317], 2*X[3233]-X[12383], X[3258]-2*X[7687], X[10295]-2*X[11657], 3*X[10706]+X[31874], 2*X[11801]-X[16340], 4*X[12068]-3*X[15035], 4*X[12900]-3*X[31378], X[16163]-2*X[22104], 3*X[23515]-2*X[31379]

 

= lies on the curves K025, Q092, Q106 and these these lines: {4,523}, {5,14385}, {30,74}, {113,14611}, {140,21315}, {230,32640}, {316,1494}, {381,9717}, {403,1300}, {477,3154}, {542,1553}, {546,3470}, {548,21317}, {671,9139}, {1263,11558}, {2777,6070}, {3233,12383}, {3258,7687}, {5523,8749}, {5962,10152}, {7471,15468}, {10113,16168}, {10295,11657}, {10297,14919}, {10706,31874}, {11801,16340}, {12068,15035}, {12900,31378}, {13202,32417}, {16163,22104}, {16243,25338}, {22265,32111}, {23515,31379}

 

= isogonal conjugate of X(15469)

 

= antigonal image of X(7471)

 

= midpoint of X(74) and X(14989)

 

= reflection of X(i) in X(j) for these {i,j}: {5,21316}, {74,12079}, {477,3154}, {3258,7687}, {7471,25641}, {10295,11657}, {12383,3233}, {14611,113}, {14934,5}, {16163,22104}, {16340,11801}

 

= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {74,5627,12079}, {477,14644,3154}, {5627,14989,74}

 

= (6-9-13)  search numbers:  [-6.1436603840234256067, -6.0605851061983525639, 10.6719897326709734659]

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CORRECTED BY Peter Moses

X(34150) =

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

= X[74] - 3 X[5627], 2 X[140] - 3 X[21315], 3 X[403] - 2 X[16319], X[477] - 3 X[14644], 2 X[3154] - 3 X[14644], 3 X[5627] - 2 X[12079], 3 X[5627] + X[14989], 3 X[9140] - X[14508], 3 X[10706] + X[31874], 4 X[12068] - 3 X[15035], 2 X[12079] + X[14989], 4 X[12900] - 3 X[31378], X[14934] - 4 X[21316], 3 X[23515] - 2 X[31379].

= lies on the cubic K025 and these lines: {4, 523}, {5, 14385}, {30, 74}, {113, 14611}, {140, 21315}, {230, 32640}, {316, 1494}, {381, 9717}, {403, 1300}, {477, 3154}, {542, 1553}, {546, 3470}, {548, 21317}, {671, 9139}, {1263, 11558}, {2777, 6070}, {3233, 12383}, {3258, 7687}, {5523, 8749}, {5962, 10152}, {7471, 15468}, {10113, 16168}, {10295, 11657}, {10297, 14919}, {10706, 31874}, {11801, 16340}, {12068, 15035}, {12900, 31378}, {13202, 32417}, {16163, 22104}, {16243, 25338}, {22265, 32111}, {23515, 31379}

= midpoint of X(i) and X(j) for these {i,j}: {74, 14989}, {476, 10733}

= reflection of X(i) in X(j) for these {i,j}: {5, 21316}, {74, 12079}, {477, 3154}, {3258, 7687}, {7471, 25641}, {10295, 11657}, {12383, 3233}, {14611, 113}, {14934, 5}, {16163, 22104}, {16340, 11801}, {21317, 548}

= isogonal conjugate of X(15469)

= antigonal image of X(7471)

= symgonal image of X(3154)

= X(1)-isoconjugate of X(15469)

= barycentric product X(i)*X(j) for these {i,j}: {94, 15468}, {1494, 3018}, {2394, 7471}, {16080, 17702}

= barycentric quotient X (i)/X(j) for these {i,j}: {6, 15469}, {2433, 15453}, {3018, 30}, {7471, 2407}, {8749, 32710}, {15468, 323}, {17702, 11064}

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {74, 5627, 12079}, {477, 14644, 3154}, {5627, 14989, 74}

Best regards,
Peter Moses.