[Antreas P. Hatzipolakis]:
Let ABC be a triangle and A'B'C' the pedal triangle of H.
Denote:
Ba, Ca = the reflections of B', C' in AH.
Ba, Ca = the reflections of B', C' in AH.
Cb, Ab = the reflections of C', A' in BH.
Ac, Bc = the reflections of A', B' in CH.
La, Lb, Lc = the Euler lines of HBaCa, HCbAb, HAcBc, resp.
La, Lb, Lc = the Euler lines of HBaCa, HCbAb, HAcBc, resp.
A*B*C* = the triangle bounded by La,Lb, Lc.
(Na), (Nb), (Nc) = the NPCs of HBaCa, HCbAb, HAcBc, resp.
1. A'B'C', A*B*C* are parallellogic.
The parallelogic center (A'B'C', A*B*C*) lies on the Euler line of ABC.
1. A'B'C', A*B*C* are parallellogic.
The parallelogic center (A'B'C', A*B*C*) lies on the Euler line of ABC.
2. The radical center of (Na), (Nb), (Nc) lies on the Euler line of ABC.
3. ABC, NaNbNc are orthologic.
The orthologic center (ABC, NaNbNc) lies on the circumcircle.
The orthologic center (ABC, NaNbNc) lies on the circumcircle.
SIMPLER:
Let ABC be a triangle and A'B'C' the pedal triangle of H.
1. A'B'C', A*B*C* are parallellogic.
Parallelogic centers?
The parallelogic center (A'B'C', A*B*C*) lies on the Euler line of ABC.
Let ABC be a triangle and A'B'C' the pedal triangle of H.
Denote:
L1, L2, L3 = the Euler lines of HB'C', HC'A', HA'B' resp.
La, Lb, Lc = the reflections of L1, L2, L3 in HA', HB', HC', resp
A*B*C* = the triangle bounded by La,Lb, Lc.
(N1), (N2), N3 = the NPCs of of HB'C', HC'A', HA'B', resp.
(Na), (Nb), (Nc) = the reflections of (N1), (N2), (N3 in HA', HB', HC', resp.
1. A'B'C', A*B*C* are parallellogic.
Parallelogic centers?
The parallelogic center (A'B'C', A*B*C*) lies on the Euler line of ABC.
2. The radical center of (Na), (Nb), (Nc) lies on the Euler line of ABC.
3. ABC, NaNbNc are orthologic.
Orthologic centers?
The orthologic center (ABC, NaNbNc) lies on the circumcircle.
Orthologic centers?
The orthologic center (ABC, NaNbNc) lies on the circumcircle.
[Peter Moses]:
Hi Antreas,
1).
(A'B'C', A*B*C*) parallelogy: X(1594).
(A*B*C*, A'B'C') parallelogy: X(11572).
2).
(a^2+b^2-c^2) (a^2-b^2+c^2) (6 a^6-7 a^4 b^2-4 a^2 b^4+5 b^6-7 a^4 c^2+16 a^2 b^2 c^2-5 b^4 c^2-4 a^2 c^4-5 b^2 c^4+5 c^6)::
on lines {{2,3},{34,10149},{1112, 6000},{1552,10421},{10152,1207 9},{12828,13202}}.
midpoint of X(382) & X(2072).
reflection of X(i) in X(j) for these {i,j}: {{468, 10151}, {10151, 4}}.
5 X[4] - X[186], 3 X[186] - 5 X[403], 3 X[4] - X[403], 4 X[186] - 5 X[468], 4 X[403] - 3 X[468], 4 X[4] - X[468], X[2071] + 3 X[3543], X[3146] + 2 X[5159], 2 X[186] - 5 X[10151], 2 X[403] - 3 X[10151], 7 X[186] - 5 X[10295], 7 X[468] - 4 X[10295], 7 X[403] - 3 X[10295], 7 X[10151] - 2 X[10295], 7 X[4] - X[10295], 2 X[3627] + X[10297], 10 X[12102] - X[12105], 3 X[13202] + X[13399].
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,378,3845),(4,382,235),(4, 1594,3861),(4,3543,25),(4, 3627,3575),(4,10736,1313),(4, 10737,1312),(4,12173,1906),( 3575,10297,468).
inverse of X(3146) in the polar circle.
X(3535)-Hirst inverse of X(3536).
3).
(ABC, NaNbNc) orthology: X(477).
(NaNbNc, ABC) orthology:
a^2 (a^6 b^2-3 a^4 b^4+3 a^2 b^6-b^8+a^6 c^2+8 a^4 b^2 c^2-3 a^2 b^4 c^2-6 b^6 c^2-3 a^4 c^4-3 a^2 b^2 c^4+14 b^4 c^4+3 a^2 c^6-6 b^2 c^6-c^8)::
on lines {{4,51},{6,12315},{20, 7998},{25,3357},{30,1216},{52, 3830},{54,12112},{64,1598},{ 140,11017},{143,12102},{373, 3855},{378,10282},{381,9729},{ 382,511},{546,5943},{548, 10170},{550,3819},{578,1498},{ 1154,13433},{1181,11403},{ 1204,10594},{1495,3520},{1503, 13403},{1593,6759},{1595,2883} ,{1596,6247},{1657,5891},{ 1872,2818},{1885,11577},{2777, 3575},{3091,11695},{3146,5562} ,{3515,11204},{3516,11202},{ 3517,10606},{3528,5650},{3529, 3917},{3543,12111},{3627, 10263},{3839,10574},{3843, 9730},{3845,5462},{3850,5892}, {3851,6688},{3853,5446},{3856, 13363},{5059,11444},{5066, 12046},{5073,10625},{5079, 10219},{5198,10605},{6152, 10628},{6995,12250},{7387, 11472},{7516,8717},{7530,7689} ,{7999,11001},{9306,12085},{ 9786,13093},{10982,12174},{ 11424,11456},{11574,12605},{ 11645,11750}}.
midpoint of X(i) and X(j) for these {i,j}: {{4, 11381}, {185, 12290}, {382, 12162}, {3146, 5562}, {5073, 10625}, {12292, 13202}}.
reflection of X(i) in X(j) for these {i,j}: {{20, 11793}, {143, 12102}, {185, 10110}, {389, 4}, {1657, 13348}, {5446, 3853}, {6241, 13382}, {10575, 9729}}.
5 X[4] - 3 X[51], 9 X[51] - 5 X[185], 3 X[4] - X[185], 6 X[51] - 5 X[389], 2 X[185] - 3 X[389], 9 X[389] - 10 X[3567], 9 X[4] - 5 X[3567], 3 X[185] - 5 X[3567], 2 X[550] - 3 X[3819], X[52] - 3 X[3830], 9 X[373] - 11 X[3855], X[3529] - 3 X[3917], 3 X[3845] - 2 X[5462], 7 X[3528] - 9 X[5650], 7 X[185] - 9 X[5890], 7 X[389] - 6 X[5890], 7 X[51] - 5 X[5890], 7 X[4] - 3 X[5890], X[1657] - 3 X[5891], 4 X[3850] - 3 X[5892], 2 X[1216] - 3 X[5907], 4 X[546] - 3 X[5943], 15 X[5890] - 7 X[6241], 5 X[185] - 3 X[6241], 5 X[389] - 2 X[6241], 5 X[4] - X[6241], 3 X[51] - X[6241], 7 X[3851] - 6 X[6688], 5 X[20] - 9 X[7998], 3 X[381] - 2 X[9729], 5 X[3843] - 3 X[9730], 11 X[389] - 14 X[9781], 11 X[4] - 7 X[9781], 9 X[5890] - 14 X[10110], 9 X[51] - 10 X[10110], 3 X[6241] - 10 X[10110], 5 X[3567] - 6 X[10110], 3 X[389] - 4 X[10110], 3 X[4] - 2 X[10110], 2 X[548] - 3 X[10170], 13 X[5079] - 12 X[10219], 3 X[3627] - X[10263], 9 X[3839] - 5 X[10574], 3 X[381] - X[10575], 7 X[1216] - 6 X[10627], 7 X[5907] - 4 X[10627], 7 X[7999] - 3 X[11001], 3 X[140] - 4 X[11017], X[389] + 2 X[11381], X[185] + 3 X[11381], 2 X[10110] + 3 X[11381], 3 X[51] + 5 X[11381], X[6241] + 5 X[11381], 3 X[5890] + 7 X[11381], 5 X[3567] + 9 X[11381], 7 X[9781] + 11 X[11381], X[20] - 5 X[11439], X[5059] - 5 X[11444], X[11381] - 3 X[11455], X[4] + 3 X[11455], X[51] + 5 X[11455], X[389] + 6 X[11455], X[5890] + 7 X[11455], X[185] + 9 X[11455], 2 X[10110] + 9 X[11455], X[6241] + 15 X[11455], 5 X[10627] - 7 X[11591], 5 X[1216] - 6 X[11591], 5 X[5907] - 4 X[11591], 5 X[3091] - 4 X[11695], 9 X[7998] - 10 X[11793], 5 X[11439] - 2 X[11793], 9 X[5943] - 8 X[12006], 3 X[546] - 2 X[12006], 9 X[5066] - 8 X[12046], 3 X[3543] + X[12111], 5 X[3091] - X[12279], 4 X[11695] - X[12279], 3 X[11381] - X[12290], 9 X[11455] - X[12290], 3 X[4] + X[12290], 2 X[10110] + X[12290], 3 X[389] + 2 X[12290], 5 X[3567] + 3 X[12290], 9 X[51] + 5 X[12290], 3 X[6241] + 5 X[12290], 9 X[5890] + 7 X[12290], 3 X[5891] - 2 X[13348], 4 X[3856] - 3 X[13363], 15 X[5890] - 14 X[13382], 5 X[185] - 6 X[13382], 5 X[389] - 4 X[13382], 5 X[10110] - 3 X[13382], 5 X[4] - 2 X[13382], 3 X[51] - 2 X[13382], 5 X[11381] + 2 X[13382], 15 X[11455] + 2 X[13382], 5 X[12290] + 6 X[13382].
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,185,10110),(4,6241,51),(4, 11455,11381),(4,12290,185),( 51,6241,13382),(51,13382,389), (64,1598,11438),(185,10110, 389),(185,11381,12290),(381, 10575,9729),(1498,1597,578),( 1593,6759,11430),(1598,3426, 64),(1657,5891,13348).
crosssum of X(3) and X(550).
Best regards,
Peter Moses.
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