HYACINTHOS 26985

[Antreas P. Hatzipolakis]:

Let ABC be a triangle and A'B'C' the cevian triangle of H.

Denote:

A"B"C" = the midheight triangle 
(ie A", B", C" = the midpoints of AA', BB', CC', resp.)

A*, B*, C* = the reflections of A", B", C" in H, resp.

Ab, Ac = the orthogonal projections of A* on AB, AC, resp.
Bc, Ba = the orthogonal projections of B* on BC, BA, resp.

Ca, Cb = the orthogonal projections of C* on CA, ab, resp.

 

La, Lb, Lc = The Euler lines of AAbAc, BBcBa, CCaCb, resp.

 

L1, L2, L3 = The Euler lines of A*AbAc, B*BcBa, C*CaCb, resp.

 

1. La, Lb, Lc are concurrent.

2. The parallels to La, Lb, Lc through A, B, C, resp. concur at X(74)

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*, resp. are concurrent

5. L1, L2, L3 are concurrent.

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

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

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

 

 

[Angel Montesdeoca]:

1. La, Lb, Lc are concurrent at

W1 =  a^2 (-a^12 (b^2+c^2)
    +4 a^10 (b^4+c^4)
    -5 a^8 (b^6+c^6)
    +4 a^6 (2 b^6 c^2-3 b^4 c^4+2 b^2 c^6)   
    +a^4 (b^2-c^2)^2 (5 b^6+b^4 c^2+b^2 c^4+5 c^6)
    +(b^2-c^2)^4 (b^6+6 b^4 c^2+6 b^2 c^4+c^6)   
    -4 a^2 (b^12-4 b^8 c^4+6 b^6 c^6-4 b^4 c^8+c^12)) : .... : ....

W1 is the midpoint of X(i) and X(j), for these {i, j}: {74,12292}, {265,7723}, {1539,15101}, {1986,12281}, {3448,12825}, {10990,11381}.
 
W1 is the reflection of X(i) in X(j), for these {i, j}: {974,125}, {1112,7687}, {1986,11746}, {11561,15088}, {11562,9826}, {12236,11801}, {13148,389}.

W1 lies on lines X(i)X(j) for these {i, j}: {4, 67}, {5, 113}, {24, 64}, {51, 14448}, {68, 265}, {110, 7503}, {146, 7544}, {184, 5609}, {186, 15138}, {389, 12099}, {399, 13198}, {468, 6000}, {542, 5907}, {973, 1112}, {1154, 13851}, {1181, 5622}, {1204, 12106}, {1216, 12358}, {1498, 5621}, {1539, 15101}, {1593, 15106}, {1658, 12041}, {1986, 3567}, {1995, 10605}, {2493, 3269}, {2777, 3575}, {2854, 11459}, {3060, 15044}, {3091, 12824}, {3331, 11062}, {3448, 12825}, {3541, 15131}, {5504, 12301}, {5562, 14984}, {5655, 14787}, {5656, 7505}, {6334, 9517}, {6639, 15061}, {6699, 7542}, {7506, 10620}, {7526, 15132}, {7569, 15102}, {7722, 15081}, {9140, 12111}, {9934, 13171}, {10113, 10263}, {10297, 13754}, {10990, 11381}, {11064, 15115}, {11801, 12236}, {12270, 15059}, {12279, 15021}, {13367, 14128}, {15025, 15043}, {15057, 15072}.

 (6 - 9 - 13) - search numbers  of W1: (-0.622150232464433, -1.67731640187864, 5.08902979049933)
 
2. The parallels to La, Lb, Lc through A, B, C, resp. concur at X(74)

3. The parallels to La, Lb, Lc through A', B', C', resp. are concurrent at X(1986).

4. The parallels to La, Lb, Lc through A*, B*, C*, resp. are concurrent at X(12292).

5. L1, L2, L3 are concurrent at  the midpoint of X(54) and X(12300):

W5 =  a^2 (-a^12 (b^2+c^2)
        +4 a^10 (b^4+b^2 c^2+c^4)
        -5 a^8 (b^6+2 b^4 c^2+2 b^2 c^4+c^6)
        +4 a^6 (4 b^6 c^2+b^4 c^4+4 b^2 c^6)       
        +a^4 (b^2-c^2)^2 (5 b^6-3 b^4 c^2-3 b^2 c^4+5 c^6)       
        -4 a^2 (b^12-b^10 c^2-b^2 c^10+c^12)
        +(b^2-c^2)^4 (b^6+4 b^4 c^2+4 b^2 c^4+c^6)) : ... : ...
       
W5 is the reflection of X(i) in X(j), for these {i, j}: {973,3574}, {6152,11743}.

W5 lies on lines X(i)X(j) for these {i, j}: {4, 9973}, {5, 51}, {54, 64}, {195, 12164}, {974, 10628}, {1199, 15140}, {1885, 11577}, {5102, 10982}, {5462, 12358}, {5876, 11424}, {5907, 5965}, {6152, 11743}, {6689, 10257}, {7691, 7998}, {10019, 11808}, {10151, 11576}, {10610, 10984}, {15058, 15069}

 (6 - 9 - 13) - search numbers  of W5: (-15.0114114533053, 5.51324857214717, 6.75214383348495)

6. The parallels to L1, L2, L3 through A, B, C, resp. are concurrent at X(54).

7. The parallels to L1, L2, L3 through A', B', C', resp. are concurrent at X(6152).

8. The parallels to L1, L2, L3 through A*, B*, C*, resp. are concurrent at X(12300).

Angel Montesdeoca