HYACINTHOS 28452

[Antreas P. Hatzipolakis]:

 

 

 

Let ABC be a triangle, A'B'C' the pedal triangle of O and L the Euler line.

 

1. Denote:

 

La, Lb, Lc = the reflections of L in BC, CA, AB, resp. concurrent at X(110)

 

L'a, L'b, L'c = the reflections of La, Lb, Lc in B'C', C'A', A'B', resp.

 

L"a, L"b, L"c = the reflections of L'a, L'b, L'c in BC, CA, AB, resp.

 

L"a, L"b, L"c  are concurrent at a point U

 

2. Denote:

 

L1, L2, L3 = the reflections of L in B'C', C'A', A'B', resp. concurrent at X(125)

 

L'1, L'2, L'3 = the reflections of L1, L2, L3 in BC, CA, AB, resp.

 

L"1, L"2, L"3 = the reflections of L'1, L'2, L'3 in B'C', C'A', A'B', resp.

 

L"1, L"2, L"3  are concurrent at a point W  

 

U = reflection of X(125) in X(110)

W = reflection of X(110) in X(125)

 

I think U, W are not listed in ETC. Are they ?

 

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[Ercole Suppa]

  

*** (1)

 

U = MIDPOINT OF X(12121) AND X(12308) 

 

Barycentrics 4 a^6 - 4 a^4 (b^2 + c^2) - (b^2 - c^2)^2 (b^2 + c^2) + a^2 (b^2 + c^2)^2 : : 

 

=X[4]-2*X[6053], 5*X[74]-7*X[3528], 5*X[265]-7*X[3851], Z*X[1511]-4*X[3530], 5*X[2948]-X[3632], 2*X[3233]-X[6070],17*X[3544]-15*X[14644], 4*X[3631]-5*X[5181], 4*X[3636]-5*X[11720], 

11*X[3855]-10*X[7687], 13*X[5079]-15*X[14643], 3*X[5655]-X[12902], 4*X[6329]-5*X[6593], 10*X[6699]-11*X[15720], 5*X[7984]-7*X[20057], 5*X[10264]-7*X[14869], 13*X[10299]-5*X[12317], 

5*X[10620]-9*X[15688], X[11008]-5*X[11061], 27*X[11693]-28*X[22250], 6*X[11737]-5*X[11801], 2*X[11800]-3*X[12824], 5*X[12295]-6*X[15687], 5*X[13605]-7*X[15808], 35*X[15036]-33*X[15715], 

7*X[15039]-4*X[20397], 25*X[15040]-21*X[15700], 5*X[15303]-4*X[20583]

 

= lies on these lines : {2,98},{4,6053},{23,5965},{51,1353},{74,3528},{113,137},{115,20976},{155,382},{159,2930},{265,3851},{373,8550},{539,10540},{541,11820},{550,5562},{1112,1843},{1495,3564},{1503,3292},{1511,3530},{2502,6388},{2777,3529},{2931,9908},{2948,3632},{3124,5477},{3167,11550},{3233,6070},{3544,14644},{3631,5181},{3636,11720},{3855,7687},{4576,14928},{5079,14643},{5191,14981},{5448,18430},{5655,12902},{5907,10619},{6030,15108},{6154,8674},{6329,6593},{6699,15720},{6791,20998},{7706,18445},{7984,20057},{9155,10991},{10116,18350},{10264,14869},{10299,12317},{10620,15688},{11008,11061},{11225,13595},{11422,19130},{11441,21659},{11693,22250},{11737,11801},{11800,12824},{12295,15687},{12310,20850},{12412,22109},{13366,18583},{13417,14984},{13431,14449},{13605,15808},{14389,18553},{15036,15715},{15039,20397},{15040,15700},{15303,20583}

 

= midpoint of X(i) and X(j) for these {i,j}: {12121,12308},{12383,14094}

 

= reflection of X(i) in X(j) for these {i,j}: {4,6053},{113,5609},{125,110},{265,16534},{3448,5972},{6070,3233},{10990,16163},{12317,20417},{15063,399},{16003,1511}

 

= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {110,125,5642},{110,3448,5972},{3448,5972,125},{6723,9140,125},{9143,14683,110},{12317,15035,20417}

 

= (6-8-13) search numbers [-1.67514575347885066, -4.88121677334253722, 7.79311259582714364]

 

 

*** (2) 

 

W = X(3448) 

 

 

Best regards

Ercole Suppa