[Antreas P. Hatzipolakis]:
Let ABC be a triangle and MaMbMc the pedal triangle of O.
Denote:
A', B', C' = the midpoints of AO, BO, CO, resp.
Na, Nb, Nc = the NPC centers of OBC, OCA, OAB, resp.
A"B"C" = the orthic triangle of A'B'C'
1. NaNbNc, A"B"C" are perspective.
2. The parallels to NaA", NbB", NcC" through A, B, C, resp. are concurrent.
3. The parallels to NaA", NbB", NcC" through A', B', C', resp. are concurrent.
4. The parallels to NaA", NbB", NcC" through Ma, Mb, Mc, resp. are concurrent.
[Ercole Suppa]:
Hi Antreas,
1. The perspector of NaNbNc, A"B"C" is
Q1 = MIDPOINT OF X(3) AND X(2904)
= a^2 (a^14-4 a^12 b^2+5 a^10 b^4-5 a^6 b^8+4 a^4 b^10-a^2 b^12-4 a^12 c^2+10 a^10 b^2 c^2-7 a^8 b^4 c^2+2 a^6 b^6 c^2-4 a^4 b^8 c^2+4 a^2 b^10 c^2-b^12 c^2+5 a^10 c^4-7 a^8 b^2 c^4+2 a^6 b^4 c^4+2 a^4 b^6 c^4-5 a^2 b^8 c^4+3 b^10 c^4+2 a^6 b^2 c^6+2 a^4 b^4 c^6+4 a^2 b^6 c^6-2 b^8 c^6-5 a^6 c^8-4 a^4 b^2 c^8-5 a^2 b^4 c^8-2 b^6 c^8+4 a^4 c^10+4 a^2 b^2 c^10+3 b^4 c^10-a^2 c^12-b^2 c^12) : : (barys)
= (11 R^4-4 R^2 SB-4 R^2 SC-7 R^2 SW+SB SW+SC SW+SW^2)S^2 -5 R^4 SB SC+5 R^2 SB SC SW-SB SC SW^2 :: (barys)
= lies on these lines: {2,24572}, {3,1986}, {5,156}, {49,18912}, {54,6644}, {70,13353}, {110,11704}, {389,12038}, {575,32366}, {578,7706}, {1147,32358}, {1614,18504}, {5012,6241}, {5449,5972}, {5944,15074}, {6642,15047}, {6689,14076}, {6759,23323}, {8538,19154}, {8548,15462}, {9826,32171}, {12827,18356}, {13198,32139}, {18570,32392}
= midpoint of X(i) and X(j) for these {i,j}: {3,2904}, {8907,15317}
= (6-9-13) search numbers: [2.4100660434271025077, 2.8879276170111470728, 0.5289918039333838184]
-------------------------------------------------------------------------
2. The parallels to NaA", NbB", NcC" through A, B, C, resp. concur at point
Q2 = X(70)
-------------------------------------------------------------------------
3. The parallels to NaA", NbB", NcC" through A', B', C', concur at point
Q3 = MIDPOINT OF X(3) AND X(70)
= a^14 b^2-5 a^12 b^4+9 a^10 b^6-5 a^8 b^8-5 a^6 b^10+9 a^4 b^12-5 a^2 b^14+b^16+a^14 c^2-2 a^12 b^2 c^2+3 a^10 b^4 c^2-4 a^8 b^6 c^2+3 a^6 b^8 c^2-6 a^4 b^10 c^2+9 a^2 b^12 c^2-4 b^14 c^2-5 a^12 c^4+3 a^10 b^2 c^4-6 a^8 b^4 c^4+6 a^6 b^6 c^4-5 a^4 b^8 c^4+3 a^2 b^10 c^4+4 b^12 c^4+9 a^10 c^6-4 a^8 b^2 c^6+6 a^6 b^4 c^6+4 a^4 b^6 c^6-7 a^2 b^8 c^6+4 b^10 c^6-5 a^8 c^8+3 a^6 b^2 c^8-5 a^4 b^4 c^8-7 a^2 b^6 c^8-10 b^8 c^8-5 a^6 c^10-6 a^4 b^2 c^10+3 a^2 b^4 c^10+4 b^6 c^10+9 a^4 c^12+9 a^2 b^2 c^12+4 b^4 c^12-5 a^2 c^14-4 b^2 c^14+c^16 : : (barys)
= (11 R^4+3 R^2 SB+3 R^2 SC-11 R^2 SW-SB SW-SC SW+2 SW^2)S^2 -5 R^4 SB SC+3 R^2 SB SC SW : : (barys)
= lies on these lines: {3,70}, {52,125}, {141,5944}, {5895,11472}, {6101,12359}, {6240,20127}, {6247,12041}, {7568,9306}, {15317,16266}, {18281,19360}
= midpoint of X(3) and X(70)
= (6-9-13) search numbers: [6.2517159543339531401, 4.9789066106789534499, -2.6916782274859516576]
----------------------------------------------------------------------------
4. The parallels to NaA", NbB", NcC" through Ma, Mb, Mc, resp. concur at point
Q4 = COMPLEMENT OF X(70)
= a^4 (a^4-2 a^2 b^2+b^4-2 a^2 c^2+c^4) (a^8-2 a^6 b^2+2 a^2 b^6-b^8-2 a^6 c^2+2 b^6 c^2-2 b^4 c^4+2 a^2 c^6+2 b^2 c^6-c^8) : : (barys)
= (10 R^4-3 R^2 SB-3 R^2 SC-7 R^2 SW+SB SW+SC SW+SW^2)S^2 -2 R^4 SB SC+3 R^2 SB SC SW-SB SC SW^2 : : (barys)
= 3*X[2]-X[70]
= lies on these lines: {2,70}, {3,6293}, {5,156}, {6,49}, {24,52}, {54,7544}, {68,110}, {113,6759}, {141,7542}, {206,9967}, {343,10020}, {578,18428}, {960,24301}, {973,1493}, {1092,1511}, {1209,6639}, {2883,12605}, {2917,11597}, {3548,15116}, {3575,13352}, {4550,7503}, {6145,10255}, {6515,11271}, {6593,8538}, {7547,15432}, {7689,11562}, {8780,9704}, {9714,18374}, {10316,11672}, {10897,10962}, {10898,10960}, {10984,13491}, {12363,32391}, {15136,15750}, {17834,22115}, {18377,26883}, {18531,32379}
= midpoint of X(2904) and X(8907)
= complement of X(70)
= complementary conjugate of X(13371)
= barycentric product of X(i) and X(j) for these {i,j}: {26,1993}, {8746,9723}
= barycentric quotient of X(i) and X(j) for these {i,j}: {26,5392}, {571,70}, {1993,20564}, {8746,847}
= trilinear product X(26)*X(47)
= trilinear quotient of X(i) and X(j) for these {i,j}: {26,91}, {571,2158}
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {569,10539,18474}
= (6-9-13) search numbers: [1.0835186814972969889, 0.5979376451754663553, 2.7266221053256884008]
Best regards
Ercole Suppa
Δεν υπάρχουν σχόλια:
Δημοσίευση σχολίου