[Antreas P; Hatzipolakis]:
Let ABC be a triangle, P a point and A'B'C' the pedal triangle of P.
Denote:
Ab, Ac = the reflections of B, C in A', resp. (PAbAc = the reflection of PBC in PA')
(Nab), (Nac) = the NPC centers of PCAb, PBAc, resp.
Bc, Ba = the reflections of C, A in B', resp.
(Nbc), (Nba) = the NPC centers of PABc, PCBa, resp.
Ca, Cb = the reflections of A, B in C', resp.
(Nca), (Ncb) = the NPC centers of PBCa, PACb, resp.
Ra, Rb, Rc = the radical axes of ((Nba), (Nca)), ((Ncb), (Nab)), ((Nac), (Nbc)), resp.
R1, R2, R3 = the parallels to Ra, Rb, Rc through A, B, C, resp.
Sa, Sb, Sc = the radical axes of ((Nbc), (Ncb)), ((Nca), (Nac)), ((Nab), (Nba)), resp.
S1, S2, S3 = the parallels to Sa, Sb, Sc through A, B, C, resp.
Which is the locus of P such that
1. Ra, Rb, Rc
2. R1, R2, R3
3. Sa, Sb, Sc
4. S1, S2, S3
are concurrent?
The entire plane?
Concurrence points?
APH
--------------------------------------------------------------------------------------------
[Ercole Suppa]
Let (x:y:z) be the barycentric coordinates of point P. We have
1. The locus of P such that Ra,Rb,Rc are concurrent is the entire plane. Concurrency point:
** Q=Q(P) = x (a^8 y z (2 x+y+z)-a^6 (b^2 z (7 x y+3 y^2+2 x z+5 y z+z^2)+c^2 y (2 x y+y^2+7 x z+5 y z+3 z^2))+(b^2-c^2)^2 x (b^4 y z+c^4 y z-b^2 c^2 (2 y z+x (y+z)))+a^2 (-b^6 z (5 x y+y^2+2 x z+3 y z+z^2)-c^6 y (2 x y+y^2+5 x z+3 y z+z^2)+b^2 c^4 (3 x y (2 y+3 z)+x^2 (5 y+3 z)+y^2 (y+4 z))+b^4 c^2 (z^2 (4 y+z)+3 x z (3 y+2 z)+x^2 (3 y+5 z)))+a^4 (b^4 z (9 x y+3 y^2+4 x z+7 y z+2 z^2)+c^4 y (4 x y+2 y^2+9 x z+7 y z+3 z^2)+b^2 c^2 (y^3+7 y^2 z+7 y z^2+z^3-2 x^2 (y+z)+2 x (y^2+3 y z+z^2)))) : : (barys)
** Pairs (P=X(i),Q=X(j)) for thes {i,j} : {1, 2646}, {3, 3}, {4, 3574}, {13, 396}, {14, 395}, {74, 21663}
** Some points:
Q1=Q(X(2))= X(549)X(29959) ∩ X(21248)X(22110)
= 4 a^8-18 a^6 b^2+25 a^4 b^4-12 a^2 b^6+b^8-18 a^6 c^2+22 a^4 b^2 c^2+28 a^2 b^4 c^2-6 b^6 c^2+25 a^4 c^4+28 a^2 b^2 c^4+10 b^4 c^4-12 a^2 c^6-6 b^2 c^6+c^8 : : (barys)
= 15 S^2-18 R^2 SB-18 R^2 SC-9 SB SC+24 R^2 SW+3 SB SW+3 SC SW+2 SW^2 : : (barys)
= on lines X(i)X(j) for these {i,j}: {549,29959},{21248,22110}
= ETC search numbers: [3.33527318943349225,2.09648328916546368,0.649896117362061673]
Q2=Q(X(5))= X(5)X(10216) ∩ X(140)X(6153)
= -2 (a-b-c)^2 (a+b-c)^2 (a-b+c)^2 (a+b+c)^2 (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (a^12-6 a^10 b^2+14 a^8 b^4-16 a^6 b^6+9 a^4 b^8-2 a^2 b^10-6 a^10 c^2+18 a^8 b^2 c^2-10 a^6 b^4 c^2-11 a^4 b^6 c^2+10 a^2 b^8 c^2-b^10 c^2+14 a^8 c^4-10 a^6 b^2 c^4-11 a^4 b^4 c^4-8 a^2 b^6 c^4+4 b^8 c^4-16 a^6 c^6-11 a^4 b^2 c^6-8 a^2 b^4 c^6-6 b^6 c^6+9 a^4 c^8+10 a^2 b^2 c^8+4 b^4 c^8-2 a^2 c^10-b^2 c^10) : : (barys)
= 3 S^4 + (-5 R^4-9 R^2 SB-9 R^2 SC-SB SC+R^2 SW+2 SB SW+2 SC SW+SW^2)S^2 + (-R^4-R^2 SW+SW^2)SB SC : : (barys)
= on lines X(i)X(j) for these {i,j}: {5,10216},{140,6153},{523,3628},{1209,13856},{9827,13467}
= ETC search numbers: [7.72264066337248837, -4.41975655759936195, 3.13620025407355812]
Q3=Q(X(6))= X(182)X(1992) ∩ X(184)X(7736)
= 2 a^4 b^2 c^2 (a^8-5 a^6 b^2+5 a^4 b^4-a^2 b^6-5 a^6 c^2+7 a^4 b^2 c^2+15 a^2 b^4 c^2-3 b^6 c^2+5 a^4 c^4+15 a^2 b^2 c^4+6 b^4 c^4-a^2 c^6-3 b^2 c^6) : : (barys)
= (3 SB+3 SC+3 SW)S^4 + (12 R^2 SB SW+12 R^2 SC SW-3 SB SC SW-4 R^2 SW^2+SB SW^2+SC SW^2+SW^3) S^2 -SB SC SW^6 : : (barys)
= on lines X(i)X(j) for these {i,j}: {182,1992},{184,7736},{575,4558},{7709,15033}
= ETC search numbers: [1.78154016551868328, 2.54580042859538244, 1.05593795494818360]
2. The locus of P such that R1,R2,R3 are concurrent is the entire plane. Concurrency point:
** Q=Q(P) = a^2 x (b^8 z^2 (y^2-x z)+(a^2-c^2)^2 y^2 (a^4 z^2-a^2 c^2 z (y+2 z)+c^4 (-x y+z^2))+b^6 z (-a^2 z (4 y^2-2 x z+y z)+c^2 (x^2 z-4 y^2 z-x (y^2+y z-2 z^2)))+b^2 y (-a^6 z^2 (4 y+z)+a^4 c^2 z (2 y^2+5 y z+2 z^2-x (y+z))+c^6 (x^2 y-4 y z^2+x (2 y^2-y z-z^2))+a^2 c^4 (2 x (y+z)^2+z (2 y^2+4 y z-z^2)))+b^4 (a^4 z^2 (6 y^2-x z+2 y z)+a^2 c^2 z (2 x (y+z)^2+y (-y^2+4 y z+2 z^2))+c^4 (2 x^2 y z+6 y^2 z^2-x (y^3-2 y^2 z-2 y z^2+z^3)))) : : (barys)
** Pairs (P=X(i),Q=X(j)) for thes {i,j} : {1, 1}, {3, 3521}, {4, 54}, {195, 21975}, {3459, 54}
** Some points:
Q4=Q(X(2)) = ISOGONAL CONJUGATE OF X(3815)
= a^2 (a^4-2 a^2 b^2+b^4-3 a^2 c^2-3 b^2 c^2) (a^4-3 a^2 b^2-2 a^2 c^2-3 b^2 c^2+c^4) : : (barys)
= (2 SB+2 SC+SW)S^2 -SB SC SW + 2 R^2 SW^2 + SB SW^2 + SC SW^2 : : (barys)
= on lines X(i)X(j) for these {i,j}: {2,5034},{25,5012},{37,26639},{39,2987},{97,14965},{111,15018},{182,263},{251,1692},{575,1976},{597,1989},{694,5038},{1994,3108},{2165,3618},{2456,11673},{2963,3589},{8770,10601},{8791,14389},{9178,21460}
= ETC search numbers: [2.00899819216986864, 1.80625984506737489, 1.46294696201318019]
Q5=Q(X(5)) = X(186)X(6152) ∩ X(523)X(8254)
= -a^2 (a-b-c)^2 (a+b-c)^2 (a-b+c)^2 (a+b+c)^2 (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (a^8-4 a^6 b^2+6 a^4 b^4-4 a^2 b^6+b^8-3 a^6 c^2+3 a^4 b^2 c^2+3 a^2 b^4 c^2-3 b^6 c^2+2 a^4 c^4+3 a^2 b^2 c^4+2 b^4 c^4+a^2 c^6+b^2 c^6-c^8) (a^8-3 a^6 b^2+2 a^4 b^4+a^2 b^6-b^8-4 a^6 c^2+3 a^4 b^2 c^2+3 a^2 b^4 c^2+b^6 c^2+6 a^4 c^4+3 a^2 b^2 c^4+2 b^4 c^4-4 a^2 c^6-3 b^2 c^6+c^8) : : (barys)
= (9 R^2-SB-SC-3 SW)S^4 + (-13 R^6-43 R^4 SB-43 R^4 SC+21 R^4 SW+30 R^2 SB SW+30 R^2 SC SW-9 R^2 SW^2-5 SB SW^2-5 SC SW^2+SW^3+SB SC (-7 R^2+3 SW))S^2 + (-R^6-R^4 SW+3 R^2 SW^2-SW^3)SB SC : : (barys)
= on lines X(i)X(j) for these {i,j}: {186,6152},{523,8254},{567,15620},{1209,13856},{6288,9221},{13434,14979}
= ETC search numbers: [8.40734861177753972, -6.09169645803289084, 3.97767805510982366]
Q6=Q(X(6)) = ISOGONAL CONJUGATE OF X(15018)
= a^4 b^4 c^4 (a^4-2 a^2 b^2+b^4-5 a^2 c^2-2 b^2 c^2+c^4) (a^4-5 a^2 b^2+b^4-2 a^2 c^2-2 b^2 c^2+c^4) : : (barys)
= 7 S^2+9 R^2 SB+9 R^2 SC-3 SB SC+3 SB SW+3 SC SW : : (barys)
= on lines X(i)X(j) for these {i,j}: {2,13337},{6,5054},{37,3582},{39,1989},{50,251},{111,3815},{308,7799},{393,13351},{549,13338},{597,2987},{1383,7736},{2165,7739},{2963,5421},{3108,5306},{6128,9698},{8749,9606}
= ETC search numbers: [1.39279115604269353, 1.70702050966112572, 1.81605436473774121]
3. The locus of P such that Sa, Sb, Sc are concurrent is the entire plane. Concurrency point:
** Q=Q(P) = x (a^10 y^2 z^2 (2 x+y+z)+(b^2-c^2)^2 x^2 (c^2 y+b^2 z) (b^4 y z+c^4 y z-b^2 c^2 (2 y z+x (y+z)))-a^8 y z (c^2 y (-2 x^2+6 x z+z (4 y+3 z))+b^2 z (-2 x^2+6 x y+y (3 y+4 z)))-a^2 (c^2 y+b^2 z) (b^6 z (y z (y+z)+x z (4 y+z)+x^2 (5 y+2 z))+c^6 y (y z (y+z)+x y (y+4 z)+x^2 (2 y+5 z))-b^2 c^4 (x^3 (y+3 z)+x^2 y (2 y+5 z)+x y (y^2+7 y z-3 z^2)+y z (2 y^2+y z-z^2))-b^4 c^2 (x^3 (3 y+z)+x^2 z (5 y+2 z)+x z (-3 y^2+7 y z+z^2)+y z (-y^2+y z+2 z^2)))+a^6 (-c^4 y^2 (x^2 (2 y+7 z)+x (y^2+4 y z-6 z^2)+z (y^2-4 y z-3 z^2))-b^4 z^2 (x^2 (7 y+2 z)+y (-3 y^2-4 y z+z^2)+x (-6 y^2+4 y z+z^2))+b^2 c^2 y z (2 x^3+y^3+5 y^2 z+5 y z^2+z^3-4 x^2 (y+z)+x (3 y^2+4 y z+3 z^2)))+a^4 (c^6 y^2 (2 y^2 z-z^3+x^2 (4 y+9 z)+2 x (y^2+4 y z-z^2))+b^6 z^2 (-y^3+2 y z^2+x^2 (9 y+4 z)+x (-2 y^2+8 y z+2 z^2))-b^4 c^2 z (2 x^3 (2 y+z)+x^2 (-6 y^2+y z-2 z^2)+x (3 y^3-3 y^2 z+y z^2-z^3)+y (y^3+y^2 z-2 y z^2+z^3))-b^2 c^4 y (2 x^3 (y+2 z)+x^2 (-2 y^2+y z-6 z^2)+z (y^3-2 y^2 z+y z^2+z^3)+x (-y^3+y^2 z-3 y z^2+3 z^3)))) : : (barys)
** Pairs (P=X(i),Q=X(j)) for thes {i,j} : {3, 3}, {4, 5}, {110, 14934}
Q7=Q(X(1)) = MIDPOINT OF X(1) AND X(15446)
= a^3 b^2 (a-b-c) c^2 (a+b+c)^3 (2 a^5-a^4 b-4 a^3 b^2+2 a^2 b^3+2 a b^4-b^5-a^4 c+6 a^3 b c-2 a^2 b^2 c-5 a b^3 c+4 b^4 c-4 a^3 c^2-2 a^2 b c^2+6 a b^2 c^2-3 b^3 c^2+2 a^2 c^3-5 a b c^3-3 b^2 c^3+2 a c^4+4 b c^4-c^5) : : (barys)
= (2 SB+2 SC+SW)S^2 - SB SC SW+2 R^2 SW^2+SB SW^2+SC SW^2 : : (barys)
= on lines X(i)X(j) for these {i,j}: {1,1399},{10,2646},{11,1385},{55,3885},{214,17606},{946,1319},{1001,10394},{1071,11715},{1388,12114},{1459,24457},{1737,26287},{1837,15079},{2320,3486},{3583,11376},{3601,11525},{3646,13384},{3649,24928},{3916,5048},{5542,20323},{10543,10959},{12743,24387},{12758,26087}
= midpoint of X(i) and X(j) for these {i,j}: {1,15446}
= ETC search numbers: [1.65894229619546952, 3.11337198732789606, 0.719587430897926665]
Q8=Q(X(2)) = X(141)X(574) ∩ X(525)X(11168)
= 4 a^10-11 a^8 b^2-2 a^6 b^4+22 a^4 b^6-14 a^2 b^8+b^10-11 a^8 c^2+16 a^6 b^2 c^2+4 a^2 b^6 c^2-5 b^8 c^2-2 a^6 c^4+36 a^2 b^4 c^4+4 b^6 c^4+22 a^4 c^6+4 a^2 b^2 c^6+4 b^4 c^6-14 a^2 c^8-5 b^2 c^8+c^10 : : (barys)
= (54 R^2+3 SB+3 SC-18 SW)S^2 + (-54 R^2+12 SW)SB SC + 18 R^2 SB SW+18 R^2 SC SW-3 SB SW^2-3 SC SW^2-2 SW^3 : : (barys)
= on lines X(i)X(j) for these {i,j}: {141,574},{525,11168}
= ETC search numbers: [0.209112163763281631, 4.08339196760851902, 0.717187505672351337}]
Q9=Q(X(6)) = X(2)X(575) ∩ X(2395)X(11166)
= 2 a^4 b^4 c^4 (4 a^10-12 a^8 b^2+14 a^6 b^4-6 a^4 b^6-12 a^8 c^2-a^6 b^2 c^2+16 a^4 b^4 c^2-11 a^2 b^6 c^2-2 b^8 c^2+14 a^6 c^4+16 a^4 b^2 c^4+22 a^2 b^4 c^4+2 b^6 c^4-6 a^4 c^6-11 a^2 b^2 c^6+2 b^4 c^6-2 b^2 c^8) : : (barys)
= (-9 SB-9 SC-7 SW)S^4 + (3 SB SC SW+12 R^2 SW^2+3 SB SW^2+3 SC SW^2-3 SW^3)S^2 -SB SC SW^3 : : (barys)
= on lines X(i)X(j) for these {i,j}: {2,575},{2395,11166}
= ETC search numbers: [2.45730760924898450, 2.51527904652153415, 0.765175476200708766]
4. The locus of P such that S1, S2, S3 are concurrent is the entire plane. Concurrency point:
** Q=Q(P) = a^2 x (a^8 y^2 z^2+a^6 y z (c^2 y (x-3 z)+b^2 (x-3 y) z)-c^8 y^2 (y z+x (y+z))-b^8 z^2 (y z+x (y+z))+b^2 c^6 y (y-z) (2 y z+x (y+z))-b^6 c^2 (y-z) z (2 y z+x (y+z))+b^4 c^4 (2 x y z (y+z)+x^2 (y+z)^2-y z (y^2-4 y z+z^2))-a^4 (-b^2 c^2 y z (x^2+y^2+3 y z+z^2-x (y+z))+b^4 z^2 (y (-3 y+z)+x (3 y+z))+c^4 y^2 ((y-3 z) z+x (y+3 z)))+a^2 (b^6 z^2 (3 x y-y^2+2 x z+2 y z)+c^6 y^2 (2 x y+3 x z+2 y z-z^2)-b^4 c^2 z (x^2 (y+z)-x (y^2-y z+z^2)+y (y^2-y z+z^2))-b^2 c^4 y (x^2 (y+z)-x (y^2-y z+z^2)+z (y^2-y z+z^2)))) : : (barys)
** Pairs (P=X(i),Q=X(j)) for thes {i,j} : {1,15446},{3,3},{4,3},{5,11273},{54,252},{74,477},{98,2698},{99,2698},{100,953},{101,2724},{102,2734},{103,2724},{104,953},{105,28914},{109,2734},{110,477},{111,6093},{476,16169},{477,16169},{930,15907},{1113,74},{1114,74},{1141,15907},{1263,24772},{1292,28914},{1296,6093},{1342,1078},{1343,1078},{1379,98},{1380,98},{1381,104},{1382,104},{3417,15446},{15620,11273}
Q10=Q(X(2)) = ISOGONAL CONJUGATE OF X(7737)
= a^2 (a^4-2 a^2 b^2+b^4-3 c^4) (a^4-3 b^4-2 a^2 c^2+c^4) : : (barys)
= (12 R^2+SB+SC-4 SW)S^2 + (-18 R^2+4 SW)SB SC + 6 R^2 SB SW+6 R^2 SC SW-2 R^2 SW^2-SB SW^2-SC SW^2 : : (barys)
= on lines X(i)X(j) for these {i,j}: {3,11653},{23,14906},{183,525},{297,11185},{378,511},{574,15066},{599,6393},{1078,9289},{3455,6800},{20977,21399}
= ETC search numbers: [-2.21114446496215498, 6.41387140689122588, 0.220820184042217331]
Q11=Q(X(6)) = ISOGONAL CONJUGATE OF X(11002)
= a^4 b^4 c^4 (2 a^4-3 a^2 b^2+2 b^4-2 a^2 c^2-2 b^2 c^2) (2 a^4-2 a^2 b^2-3 a^2 c^2-2 b^2 c^2+2 c^4) : : (barys)
= 7 S^4 + (9 R^2 SB+9 R^2 SC-3 SB SC-12 R^2 SW-4 SB SW-4 SC SW+3 SW^2)S^2 + SB SC SW^2: : (barys)
= on lines X(i)X(j) for these {i,j}: {50,5094},{182,599},{183,7496},{186,2453},{187,18575},{7771,11643},{7778,10130}
= ETC search numbers: [3.92170672343229917, 3.54119535044456357, -0.620950786907533756]
Best Regards
Ercole Suppa
Δεν υπάρχουν σχόλια:
Δημοσίευση σχολίου