[Antreas P. Hatzipolakis]:
Let ABC be a triangle and P a point and A'B'C' the pedal triangle of P.
Denote:
(Na), (Nb), (Nc) = the NPCs of PBC, PCA, PAB, resp.
(Nab), (Nac) = the reflections of (Nb), (Nc) in PNa, resp.
(Nbc), (Nba) = the reflections of (Nc), (Na) in PNb, resp.
(Nca), (Ncb) = the reflections of (Na), (Nb) in PNc, resp.
Sa, Sb, Sc = the radical axes of ((Nab), (Nac)), ((Nbc), (Nba)),
((Nca), (Ncb)), resp,
Which is the locus of P such that:
1. Sa, Sb, Sc are concurrent?
2. the reflections of Sa, Sb, Sc in BC, CA, AB, resp. are concurrent?
3. the parallels to Sa, Sb, Sc through A, B, C, resp. are concurrent?
4. the parallels to Sa, Sb, Sc through A', B', C', resp. are concurrent?
I lies on the loci.
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[Ercole Suppa]
The equations of loci are very complicated.
1. The locus of P such that Sa, Sb, Sc are concurrent is a curve of order 17 through A,B,C,X(1),X(3),X(4),X(13),X(14).
Let Q1=Q1(P) be the point of concurrence.
** Pairs (P=X(i),Q1=X(j)) for thes {i,j} : {1,1317}, {3,30}, {13,530}, {14,531}
2. The locus of P such that the reflections of Sa, Sb, Sc in BC, CA, AB, resp. are concurrent is a curve of order 18 through A,B,C,X(1),X(3),X(4),X(110).
Let Q2=Q2(P) be the point of concurrence.
** Pairs (P=X(i),Q2=X(j)) for thes {i,j} : {1,1319}, {110,25641}
** Some points P in the locus with Q2(P) point of concurrence:
Q2(X(3)) = MIDPOINT OF X(3) AND X(23236)
= -4 a^10+10 a^8 (b^2+c^2)+(b^2-c^2)^4 (b^2+c^2)-7 a^6 (b^2+c^2)^2-a^2 (b^2-c^2)^2 (b^4+c^4)+a^4 (b^6+5 b^4 c^2+5 b^2 c^4+c^6) : : (barys)
= 2*X[2]-3*X[11693], X[4]-3*X[110], 2*X[5]-3*X[5642], 3*X[74]-5*X[3522], 3*X[146]+X[5059], 3*X[265]-5*X[1656], 3*X[376]-X[15054], 3*X[381]-7*X[15039], X[382]-3*X[5655], 3*X[549]-2*X[20379], 2*X[576]-3*X[15303], 5*X[631]-3*X[9140], 5*X[632]-4*X[20396], 3*X[1495]-2*X[16619], 3*X[1568]-2*X[18572], X[3146]-3*X[10706], 2*X[3233]-X[25641], 3*X[3448]-7*X[3523], 9*X[3524]-7*X[15057], 7*X[3526]-5*X[15027], 7*X[3528]-5*X[15021], 17*X[3533]-15*X[15059], 9*X[3545]-7*X[15044], 2*X[3628]-3*X[11694], 4*X[3850]-3*X[10113], 7*X[3851]-6*X[7687], 11*X[3855]-13*X[15029], 11*X[5056]-12*X[12900], 13*X[5067]-11*X[15025], X[5073]-6*X[6053], 2*X[5446]-3*X[12824], 3*X[5891]-2*X[15738], X[7722]+X[12273], 3*X[10264]-5*X[15712], 13*X[10299]-15*X[15051], 3*X[10540]-X[18325], 5*X[11522]-6*X[11723], 3*X[11562]-2*X[13148], 3*X[11597]-2*X[12242], 3*X[11702]-2*X[11803], 3*X[11720]-2*X[13464], 2*X[11800]-3*X[16222], X[11801]-2*X[13392], X[12219]+X[15102], 2*X[12236]-3*X[16223], X[12308]+X[20127], X[12317]-3*X[15055], 2*X[14156]-X[25739], 2*X[14708]-X[21649], 9*X[14845]-8*X[15465], 5*X[15040]-3*X[15061], 3*X[16164]-2*X[16617], 3*X[16165]-2*X[16618]
= on lines X(i)X(j) for these {i,j}: {2,11693},{3,67},{4,110},{5,5642},{20,541},{24,12828},{30,3292},{52,5095},{74,3522},{125,128},{146,5059},{186,539},{265,1656},{376,15054},{381,15039},{382,5655},{399,1498},{549,20379},{550,5562},{567,25555},{569,15118},{576,15303},{631,9140},{632,20396},{690,10992},{1092,11750},{1154,14448},{1495,16619},{1503,10564},{1568,18572},{1986,13431},{2771,12757},{2781,10625},{2836,12675},{2854,8550},{2929,5898},{2931,3515},{3146,10706},{3233,25641},{3448,3523},{3516,12168},{3517,12310},{3519,21394},{3524,15057},{3526,15027},{3528,15021},{3533,15059},{3545,15044},{3581,5965},{3628,11694},{3850,10113},{3851,7687},{3855,15029},{4857,12896},{5056,12900},{5067,15025},{5073,6053},{5094,15115},{5270,18968},{5446,12824},{5449,11449},{5622,13336},{5891,15738},{7495,18475},{7533,15033},{7574,15139},{7722,12273},{8542,11179},{8674,10993},{8907,12893},{9033,12790},{9517,14900},{9703,18388},{9977,15037},{10116,22467},{10264,15712},{10295,13754},{10299,15051},{10540,18325},{10620,11850},{11411,25712},{11430,18553},{11522,11723},{11561,14049},{11562,13148},{11597,12242},{11702,11803},{11720,13464},{11800,16222},{11801,13392},{12038,12827},{12134,18488},{12219,15102},{12236,16223},{12308,20127},{12317,15055},{13403,18350},{14156,25739},{14708,21649},{14805,24206},{14845,15465},{15040,15061},{15473,19504},{16164,16617},{16165,16618},{16176,17834},{18555,20771}
= midpoint of X(i) and X(j) for these {i,j}: {3,23236},{74,14683},{110,12383},{12219,15102}
= reflection of X(i) in X(j) for these {i,j}: {4,16534},{52,25711},{113,110},{125,1511},{265,5972},{3448,6699},{5181,12584},{7728,6053},{10113,10272},{11801,13392},{12295,113},{12902,7687},{15063,5609},{15133,15115},{16003,3},{21649,14708},{25641,3233},{25739,14156}
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4,110,16534},{4,16534,113},{20,9143,14094},{265,5972,23515},{550,10990,16111},{631,9140,20397},{3448,15035,6699},{9140,15020,631},{10990,16163,550},{12902,14643,7687}
= [4.19965416661580419, 1.91447334573039350, 0.376957934886812438]
3. The locus of P such that the parallels to Sa, Sb, Sc through A, B, C, resp. are concurrent is a curve of order 17 through A,B,C,X(1),X(3),X(4),X(13),X(14).
Let Q3=Q3(P) be the point of concurrence.
** Pairs (P=X(i),Q3=X(j)) for these {i,j} : {1,80}, {3,30}, {13,530}, {14,531}
4. The locus of P such that the parallels to Sa, Sb, Sc through A', B', C', resp. are concurrent is a curve of order 18 through A,B,C,X(1),X(3),X(4),X(13),X(14).
Let Q4=Q4(P) be the point of concurrence.
** Pairs (P=X(i),Q4=X(j)) for these {i,j} : {1,1317},{3,30},{13,530},{14,531}
Best regards
Ercole Suppa
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