HYACINTHOS 29258

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle and A'B'C' the pedal triangle of I..

 

Denote:

 

Na, Nb, Nc = the NPC centers of IBC, ICA, IAB, resp.

 

N1N2N3 = the reflection triangle of NaNbNc

(ie N1, N2, N3 = the reflections of Na, Nb, Nc in NbNc, NcNa, NaNb, resp.)

 

1. A'B'C', N1N2N3 are orthologic.

 

2. Which point is the circumcenter of N1N2N3?

It has an interesting property:

It is lying on the parallel through I to Euler line of ABC.

 

 

[Ercole Suppa]:

 

 

Hi Antreas, 

 

We have:

 

Z1 = orthology center(A'B'C', N1N2N3) = 

 

= REFLECTION OF X(11) IN X(10122)

 

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

 

= X[11]-2*X[10122], 3*X[21]-X[12532], 2*X[3035]-X[31938], X[3649]-2*X[5083], X[5441]+X[11571], X[12758]-2*X[15174], 3*X[15670]-2*X[18254] 

 

= lies on these lines: {1,399}, {7,11604}, {11,10122}, {21,12532}, {30,11570}, {57,13146}, {79,6583}, {214,5427}, {354,1484}, {442,20118}, {758,1317}, {952,13375}, {2800,10543}, {3035,31938}, {3649,5083}, {5441,11571}, {6917,10044}, {10052,16159}, {10087,16139}, {10427,12832}, {11263,13751}, {12758,15174}, {14151,16140}, {14795,22937}, {14804,22935}, {15670,18254}

 

= midpoint of X(i) and X(j) for these {i,j}: {5441,11571}, {17637,17660}

 

= reflection of X(i) in X(j) for these {i,j}: {11,10122}, {3649,5083}, {12758,15174}, {31938,3035}

 

= (6-9-13)  search numbers:  [0.3985968034562317795, -0.1309271404557180266, 3.5473385929354615310]

 

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Z2 = orthology center(N1N2N3,A'B'C') = X(3874) 

 

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W = circumcenter of N1N2N3 =

 

= COMPLEMENT OF X(13465)

 

= 2 a^6 b-a^5 b^2-5 a^4 b^3+2 a^3 b^4+4 a^2 b^5-a b^6-b^7+2 a^6 c+a^4 b^2 c-a^3 b^3 c-4 a^2 b^4 c+a b^5 c+b^6 c-a^5 c^2+a^4 b c^2-4 a^3 b^2 c^2+a b^4 c^2+3 b^5 c^2-5 a^4 c^3-a^3 b c^3-2 a b^3 c^3-3 b^4 c^3+2 a^3 c^4-4 a^2 b c^4+a b^2 c^4-3 b^3 c^4+4 a^2 c^5+a b c^5+3 b^2 c^5-a c^6+b c^6-c^7 : : (barys)

 

= a b c SA SB+2 a SA SB^2+6 b SA SB^2-2 c SA SB^2+a b c SA SC+2 a b c SB SC+4 b SA SB SC+4 c SA SB SC+2 a SB^2 SC+2 b SB^2 SC-2 c SB^2 SC+2 a SA SC^2-2 b SA SC^2+6 c SA SC^2+2 a SB SC^2-2 b SB SC^2+2 c SB SC^2 : : (barys)

 

= 3*X[2]-X[13465], X[3]+X[14450], 2*X[140]-X[191], 3*X[549]-2*X[22937], 5*X[631]-X[31888], 2*X[1125]-X[22936], 5*X[3616]-3*X[28453], 2*X[3647]-3*X[31650], X[3650]-3*X[28465], X[3652]-2*X[10021], 3*X[5886]-X[7701], 2*X[6675]-X[19919], 3*X[10246]-X[15680], 2*X[11277]-X[16139], 2*X[11281]-X[31649]

 

= lies on these lines: {1,30}, {2,13465}, {3,14450}, {5,2771}, {12,11571}, {104,5606}, {140,191}, {442,25005}, {495,14526}, {496,13751}, {549,22937}, {631,31888}, {758,5499}, {946,12267}, {952,2475}, {1071,22805}, {1125,22936}, {1483,13463}, {1484,12005}, {1749,5433}, {3616,28453}, {3647,31650}, {3648,5303}, {3650,28465}, {3651,11849}, {3652,10021}, {3754,11698}, {3845,18243}, {5426,12409}, {5428,17768}, {5563,18244}, {5763,26285}, {5844,16126}, {5886,7701}, {6583,12909}, {6675,19919}, {9955,12009}, {10246,15680}, {10711,18357}, {10916,24475}, {10943,17653}, {11277,16139}, {11281,31649}, {11684,27529}, {12006,31847}, {12433,22938}, {12913,22765}, {13100,16150}, {14988,24987}, {16117,28174}, {25557,26202}

 

= midpoint of X(i) and X(j) for these {i,j}: {3,14450}, {13743,16116}, {16132,16159}

 

= complement of X(13465)

 

= reflection of X(i) in X(j) for these {i,j}: {5,11263}, {191,140}, {3652,10021}, {5690,5499}, {13743,5901}, {16139,11277}, {19919,6675}, {22936,1125}, {31649,11281}

 

= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {3652,26725,10021}

 

= (6-9-13)  search numbers:  [2.6712094246583812338, 2.6686217832766554239, 0.5602912051812878556]

 

 

Best regards,

Ercole Suppa