HYACINTHOS 26206

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle.

Denote:

A', B', C' = the reflections of A, B, C in I, resp.

Ab, Ac = the orthogonal projections of A' on BI, CI, resp.

L1 = the Euler line of IAbAc. Similarly L2, L3.

(N1) = the NPC of IAbAc. Similarly (N2), (N3)

1. L1, L2, L3 are concurrent 

2. The parallels to L1, L2, L3 through A, B, C, resp. are concurrent.

3. (N1), (N2), (N3) are concurrent.

 

4. ABC, N1N2N3 are perspective

 

5. ABC, N1N2N3 are orthologic.

 

 

[Peter Moses]:

Hi Antreas,

 

1). X(10543).

2). X(79).

3). X(12735).

 

4). (a^2-5 a b+b^2-c^2) (a^2-b^2-5 a c+c^2):: = Sin[A]/(5-2 Cos[A]):: 

on lines {{1,549},{7,5697},{8,389 8},{9,3633},{21,3244},{35,1476 },{79,3057},{80,9957},{104, 3746},...}.

on Feuerbach.

isoconjugate of X(58) and X(3918).

trilinear pole of line {650, 4949, 4984}.

barycentric quotient X(37)/X(3918).

 

5). 

(ABC, NaNbNc) orthology: X(104).

 

(NaNbNc, ABC) orthology: 

6 a^4-5 a^3 b-5 a^2 b^2+5 a b^3-b^4-5 a^3 c+10 a^2 b c-5 a b^2 c-5 a^2 c^2-5 a b c^2+2 b^2 c^2+5 a c^3-c^4:: 

on lines {{1,4},{3,3244},{5,3636} ,{8,10165},{10,3526},{40,3241} ,{104,3746},...}.

Midpoint of X(i) and X(j) for these {i,j}: {{1,5882},{3,3244},{145,11362} ,{550,11278},{944,946},{1317, 11715},{1385,1483},{1482,4297} ,{3057,5884},{5493,8148},{ 7972,10265},{9957,12675}}.

Reflection of X(i) in X(j) for these {i,j}: {{5,3636},{3626,140},{3754, 13373},{6684,1385},{13464,1}}.

5 X[1] - X[4], 3 X[1] + X[944], 3 X[4] + 5 X[944], 3 X[4] - 5 X[946], 3 X[1] - X[946], 11 X[4] - 15 X[1699], 11 X[946] - 9 X[1699], 11 X[1] - 3 X[1699], 11 X[944] + 9 X[1699], X[40] + 3 X[3241], 5 X[10] - 7 X[3526], 7 X[4] - 15 X[5603], 7 X[1699] - 11 X[5603], 7 X[946] - 9 X[5603], 7 X[1] - 3 X[5603], 7 X[944] + 9 X[5603], 9 X[4] - 5 X[5691], 9 X[1] - X[5691], 3 X[946] - X[5691], 3 X[944] + X[5691], X[944] - 3 X[5882], X[946] + 3 X[5882], X[4] + 5 X[5882], 3 X[5603] + 7 X[5882], X[5691] + 9 X[5882], 3 X[1699] + 11 X[5882], X[944] - 9 X[7967], X[5882] - 3 X[7967], X[1] + 3 X[7967], X[5603] + 7 X[7967], X[946] + 9 X[7967], X[1699] + 11 X[7967], X[4] + 15 X[7967], X[8] - 3 X[10165], 7 X[3526] - 15 X[10246], X[10] - 3 X[10246], 5 X[8] - 13 X[10303], 15 X[10165] - 13 X[10303], 5 X[40] - 9 X[10304], 5 X[3241] + 3 X[10304], 9 X[1] - 5 X[10595], 3 X[946] - 5 X[10595], X[5691] - 5 X[10595], 3 X[944] + 5 X[10595], 9 X[5882] + 5 X[10595], 13 X[946] - 15 X[11522], 13 X[10595] - 9 X[11522], 13 X[1] - 5 X[11522], 13 X[5882] + 5 X[11522], 13 X[944] + 15 X[11522], 3 X[10] - X[12645], 9 X[10246] - X[12645], 10 X[11522] - 13 X[13464], 6 X[1699] - 11 X[13464], 2 X[5691] - 9 X[13464], 10 X[10595] - 9 X[13464], 6 X[5603] - 7 X[13464], 2 X[4] - 5 X[13464], 2 X[946] - 3 X[13464], 2 X[5882] + X[13464], 6 X[7967] + X[13464], 2 X[944] + 3 X[13464].

{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,944,946),(1,5691,10595),(1, 7967,5882),(944,10595,5691),( 946,5882,944),(5691,10595,946) .

 

 

Best regards,

Peter Moses.