HYACINTHOS 26760

[Antreas P. Hatzipolakis]:

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

Denote:

A"B"C" = the orthic triangle of A'B'C'

(Nab), (Nac) = the NPCs of A'BB", A'CC", resp.

(Nbc), (Nba) = the NPCs of B'CC", B'AA", resp.

(Nca), (Ncb) = the NPCs of C'AA", C'BB", resp.

R1 = the radical axis of (Nba), (Nca)
R2 = the radical axis of (Ncb), (Nab)
R3 = the radical axis of (Nac), (Nbc)

1. R1, R2, R3 are concurrent.
2. The parallels to R1, R2, R3 through A, B, C, resp. are concurrent.

3. The parallels to R1, R2, R3 through A', B', C', resp. are concurrent.

4. The parallels to R1, R2, R3 through A", B", C", resp. are concurrent.

[Angel Montesdeoca]:

 

1. R1, R2, R3 are concurrent at X(142).

2. The parallels to R1, R2, R3 through A, B, C, resp. are concurrent at X(9).

3.The parallels to R1, R2, R3 through A', B', C', resp. are concurrent at

W = (a (a^3 (b+c)-(b-c)^2 (b^2+c^2)-a^2 (3 b^2+2 b c+3 c^2)+a (3 b^3-b^2 c-b c^2+3 c^3)) : .... : ....)

W lies lies on these lines: {1,6}, {38,4343}, {57,3174}, {65,5853}, {142,354}, {144,4430}, {145,7672}, {210,6666}, {390,3868}, {480,8257}, {516,1071}, {527,3058}, {528,11570}, {664,10509}, {942,2550}, {962,12669}, {971,12699}, .....

W = midpoint of X(i) and X(j) for these {i,j}: {145,7672}, {390,3868}, {962,12669}, {1320,12755}, {3555,5728}, {4430,7671}.

W = reflection of X(i) in X(j) for these {i, j}: {9,5572}, {72,1001}, {2550,942}, {3059,142}, {5542,3881}, {5732,12675}, {5784,5542}, {10427,5083}.


 (6 - 9 - 13) - search numbers  of W: (-0.405790345817078, 1.08634782234415, 3.07586537989246).
 
 4. The parallels to R1, R2, R3 through A", B", C", resp. are concurrent at X(7).
 
 Angel Montesdeoca