[Antreas P. Hatzipolakis]:
Let ABC be a triangle and A'B'C' the pedal triangle of H.
Denote:
A"B"C" = the reflection triangle
(ie A", B", C" = the reflections of A,B,C in BC, CA, AB, resp.)
Ab, Ac = the orthogonal projections of A" on BB', CC', resp.
La = the Euler line of A"AbAc. Similarly Lb, Lc.
1. La, Lb, Lc are concurrent.
2. The parallels to La, Lb, Lc through A, B, C, resp. concur at X74
Denote:
A"B"C" = the reflection triangle
(ie A", B", C" = the reflections of A,B,C in BC, CA, AB, resp.)
Ab, Ac = the orthogonal projections of A" on BB', CC', resp.
La = the Euler line of A"AbAc. Similarly Lb, Lc.
1. La, Lb, Lc are concurrent.
2. The parallels to La, Lb, Lc through A, B, C, resp. concur at X74
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.
4. The parallels to La, Lb, Lc through A", B", C", resp are concurrent.
[Peter Moses]:
Hi Antreas,
1). a^2 (a^8 b^2-2 a^6 b^4+2 a^2 b^8-b^10+a^8 c^2+a^4 b^4 c^2-3 a^2 b^6 c^2+b^8 c^2-2 a^6 c^4+a^4 b^2 c^4+2 a^2 b^4 c^4-3 a^2 b^2 c^6+2 a^2 c^8+b^2 c^8-c^10)::
1). a^2 (a^8 b^2-2 a^6 b^4+2 a^2 b^8-b^10+a^8 c^2+a^4 b^4 c^2-3 a^2 b^6 c^2+b^8 c^2-2 a^6 c^4+a^4 b^2 c^4+2 a^2 b^4 c^4-3 a^2 b^2 c^6+2 a^2 c^8+b^2 c^8-c^10)::
on lines {{2,13201},{3,11557},{4,7730}, {6,1205},{23,110},{30,11562},{ 51,125},{52,3627},{67,9969},{ 74,389},{113,5562},{143,10264} ,{146,5889},{184,10117},{185, 1986},{265,5446},{373,6723},{ 399,6243},{550,11561},{568, 10620},{974,10990},{1181,9919} ,{1204,2935},{1498,12165},...} .
Complement X(13201).
Midpoint of X(i) and X(j) for these {i,j}: {{4,7731},{146,5889},{399, 6243},{3146,12270},{7722, 10721}}.
Reflection of X(i) in X(j) for these {i,j}: {{3,11557},{4,11807},{67,9969} ,{74,389},{125,1112},{185, 1986},{265,5446},{550,11561},{ 1205,6},{3313,6593},{3448, 11800},{3917,12824},{5562,113} ,{6101,10272},{6467,5095},{ 10264,143},{10620,11806},{ 10625,1511},{10990,974},{ 11381,13202},{12162,1539},{ 12219,5907}}.
3 X[51] - 2 X[125], 3 X[51] - 4 X[1112], 9 X[373] - 8 X[6723], 3 X[568] - X[10620], 9 X[51] - 8 X[11746], 3 X[125] - 4 X[11746], 3 X[1112] - 2 X[11746], 3 X[568] - 2 X[11806], X[7731] + 2 X[11807], 10 X[11746] - 9 X[12099], 5 X[125] - 6 X[12099], 5 X[51] - 4 X[12099], 5 X[1112] - 3 X[12099], 3 X[4] - X[12281], 6 X[11807] - X[12281], 3 X[7731] + X[12281], 5 X[110] - 8 X[13402].
Complement X(13201).
Midpoint of X(i) and X(j) for these {i,j}: {{4,7731},{146,5889},{399, 6243},{3146,12270},{7722, 10721}}.
Reflection of X(i) in X(j) for these {i,j}: {{3,11557},{4,11807},{67,9969} ,{74,389},{125,1112},{185, 1986},{265,5446},{550,11561},{ 1205,6},{3313,6593},{3448, 11800},{3917,12824},{5562,113} ,{6101,10272},{6467,5095},{ 10264,143},{10620,11806},{ 10625,1511},{10990,974},{ 11381,13202},{12162,1539},{ 12219,5907}}.
3 X[51] - 2 X[125], 3 X[51] - 4 X[1112], 9 X[373] - 8 X[6723], 3 X[568] - X[10620], 9 X[51] - 8 X[11746], 3 X[125] - 4 X[11746], 3 X[1112] - 2 X[11746], 3 X[568] - 2 X[11806], X[7731] + 2 X[11807], 10 X[11746] - 9 X[12099], 5 X[125] - 6 X[12099], 5 X[51] - 4 X[12099], 5 X[1112] - 3 X[12099], 3 X[4] - X[12281], 6 X[11807] - X[12281], 3 X[7731] + X[12281], 5 X[110] - 8 X[13402].
crosssum of X(3) and X(3448).
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (125,1112,51),(568,10620, 11806).
......
3). X(1986).
4). X(7731).
Best regards,
Peter Moses.
Best regards,
Peter Moses.
Δεν υπάρχουν σχόλια:
Δημοσίευση σχολίου