HYACINTHOS 25068

[Antreas P. Hatzipolakis]:

Let ABC be a triangle and IaIbIc the antipedal triangle of I

Denote:

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

A2, A3 = the orthogonal projections of Ia on AAb, AAc, resp.

La = the Euler line of IaA2A3. Similarly Lb, Lc

1. La, Lb, Lc concur on the Euler line of ABC.

2. The parallels to La, Lb, Lc through Ia, Ib, Ic, resp. are concurrent.

3. The parallels to La, Lb, Lc through A, B, C, resp. are concurrent.

4. The parallels to La, Lb, Lc through A', B', C' are concurrent
(where A'B'C' is the pedal triangle of I).

5. The parallels to La, Lb, Lc through A", B", C" are concurrent
(where A", B", C" are the orthogonal projections of Ia, Ib, Ic on BC, CA, AB, resp.)

ORTHIC TRIANGLE VERSION:

Let ABC be a triangle and HaHbHc the pedal triangle of H.

Denote:

Ab, Ac = the orthogonal projections of Ha on HbH, HcH, resp.
A2, A3 = the orthogonal projections of A on HaAb, HaAc, resp.

La = the Euler line of AA2A3. Similarly Lb, Lc

6. La, Lb, Lc are concurrent on the Euler line of HaHbHc
7. The parallels to La, Lb, Lc through A, B, C, resp. are concurrent.
8. The parallels to La, Lb, Lc through Ha, Hb, Hc, resp. are concurrent.

9. The parallels to La, Lb, Lc through A', B', C' are concurrent
(where A'B'C' is the pedal triangle of H wrt triangle HaHbHc).

10. The parallels to La, Lb, Lc through A", B", C" are concurrent
(where A", B", C" are the orthogonal projections of A, B, C on HbHc, HcHa, HaHb, resp.)

 

[Angel Montesdeoca]:

 1. La, Lb, Lc concur at X(442) = complement of  Schifler Point, on the Euler line of ABC.
2. The parallels to La, Lb, Lc through Ia, Ib, Ic, resp. are concurrent at X(191) = reflection of incenter in  Schifler Point.
3. The parallels to La, Lb, Lc through A, B, C, resp. are concurrent at X(79).
4. The parallels to La, Lb, Lc through A', B', C' are concurrent at X(9850)  = orthologic center of these triangles: intouch to andromeda.
    (where A'B'C' is the pedal triangle of I).
 5. The parallels to La, Lb, Lc through A", B", C" are concurrent at X(3650) = = KS(extouch triangle)
    (where A", B", C" are the orthogonal projections of Ia, Ib, Ic on BC, CA, AB, resp.; A"B"C"=extouch triangle)
   
    6. La, Lb, Lc are concurrent at X(973) = 1st Ehrmann Point,  on the Euler line of HaHbHc
    7. The parallels to La, Lb, Lc through A, B, C, resp. are concurrent at X(54) = Kosnita Point
    8. The parallels to La, Lb, Lc through Ha, Hb, Hc, resp. are concurrent at X(6152)  = orthic-triangle-orthologic center of reflection triangle
     9. The parallels to La, Lb, Lc through A', B', C'   (where A'B'C' is the pedal triangle of H wrt triangle HaHbHc) are concurrent at  U = X(4) + X(6152), the midpoint of X(4) and X(6152) ; U =2 X(3) - X(9827), the  reflection of X(3) in X(9827);
    
     U = (a^2 (a^12 (b^2+c^2)-4 a^10 (b^2+c^2)^2+5 a^8 (b^6+2 b^4 c^2+2 b^2 c^4+c^6)+2 a^6 b^2 c^2 (b^2-c^2)^2+-a^4 (b^2-c^2)^2 (5 b^6+11 b^4 c^2+11 b^2 c^4+5 c^6)+2 a^2 (b^2-c^2)^4 (2 b^4+3 b^2 c^2+2 c^4)-(b^2-c^2)^4 (b^6-2 b^4 c^2-2 b^2 c^4+c^6)) : ... : ...),
    
     with (6-9-13)-search numbers      (3.40260281068115,  -9.93280952522067,  8.94679285597662).
    U lie on lines:  {3, 9827}, {4, 93}, {25, 54}, {66, 6145}, {143, 7576}, {156, 1493}, {195, 1598}, {235, 1843}, {389, 973}, {427, 1209}, {428, 539}, {468, 6689}, {567, 3518}, {1112, 6756}, {1351, 5198}, {1593, 7691}, {1885, 6153}, {2914, 5609}, {6241, 7730}, {6403, 7507}, {7713, 9905}.
    
  Note:  The points X(442),  X(191),  X(79) and X(3650)  are aligned.
 
    10. The parallels to La, Lb, Lc through A", B", C"      (where A", B", C" are the orthogonal projections of A, B, C on HbHc, HcHa, HaHb, resp.) are concurrent at V= 2 X(973) - X(54), the  reflection of X(973) in X(54);

V=  (a^2 (-a^12 (b^2+c^2)  +4 a^10 (b^4+3 b^2 c^2+c^4)-a^8 (5 b^6+26 b^4 c^2+26 b^2 c^4+5 c^6)
+20 a^6 b^2 c^2 (b^4+b^2 c^2+c^4)+a^4 (b^2-c^2)^2 (5 b^6+b^4 c^2+b^2 c^4+5 c^6)-4 a^2 (b^2-c^2)^2 (b^8+b^4 c^4+c^8)+(b^2-c^2)^4 (b^6+c^6)) : ... : ...),

with (6-9-13)-search numbers (-15.5225510948475,  22.7987837399416,  -4.97885452504560).
V lie on lines: {6,24}, {25,9935}, {184,1493}, {185,550}, {539,1216}, {1596,3574}, {1899,3519}, {4549,9936}.

Note:  The points X(973), X(54), X(6152)  and V are aligned.

Angel Montesdeoca