Let ABC be a triangle, P a point and A'B'C' the pedal triangle of P.
Denote:
Ab, Ac = the orthogonal projections of A on PB, PC, resp.
Bc, Ba = the orthogonal projections of B on PC, PA, resp.
Ca, Cb = the orthogonal projections of C on PA, PB, resp.
A*B*C* = the triangle bounded by AbAc, BcBa, CaCb
La =: PA*, Lb =: PB*, Lc =: PC*
Which is the locus of P such that:
1. The parallels to La, Lb, Lc through A, B, C, resp. are concurrent ?
2. The parallels to La, Lb, Lc through A', B', C', resp. are concurrent ?
O lies on the loci.
Points of concurrences?
[César Lozada]:
1)
Locus={Linf} ∪ {circumcircle} ∪ {K006 (orthocubic)}
If P lies on the circumcircle then the point of intersection of the given lines Q1(P) = HP ∩ {Linf}
ETC pairs (P ∈ circumcircle,Q1(P)): (74, 2777), (98, 2794), (100, 5840), (104, 2829), (107, 2777), (108, 2829), (110, 17702), (112, 2794) , (915,5840), (1113,30), (1114,30), (1300,17702)
K006 passes through ETC’s: 1,3,4,46,90,155,254,371,372,485,486,487,488,6212,6213,8946,8947,8948,8949
ETC pairs (P ∈ K006,Q1(P)): (1,8), (3,22261), (4,4)
P01 = Q1 (X(46) ) = X(63)X(499) ∩ X(155)X(2990)
= (b+c-a)*(a^3+(b+c)*a^2-(b^2+c^2)*a-(b^2-c^2)*(b-c))*(a^6+2*c*a^5-(3*b^2+c^2)*a^4-4*c*(b^2-b*c+c^2)*a^3+(3*b^4-c^4-2*b*c^2*(b-2*c))*a^2+2*(b^2-c^2)^2*c*a-(b^2-c^2)^3)*(a^6+2*b*a^5-(b^2+3*c^2)*a^4-4*b*(b^2-b*c+c^2)*a^3-(b^4-3*c^4-2*b^2*c*(2*b-c))*a^2+2*(b^2-c^2)^2*b*a+(b^2-c^2)^3) : : (barys)
= on lines: {63, 499}, {155, 2990}, {912, 6504}, {6505, 10052}
= [ 8.7100558299150650, 4.2477158056617920, -3.3200876127422770 ]
P02 = Q1 ( X(99) ) = INFINITY POINT OF THE LINE X(4)X(99)
= 2*a^8-4*(b^2+c^2)*a^6+(3*b^4+4*b^2*c^2+3*c^4)*a^4-2*(b^2+c^2)*b^2*c^2*a^2-(b^2-c^2)^4 : : (barys)
= on lines: {2, 9734}, {3, 115}, {4, 99}, {5, 620}, {12, 15452}, {13, 5474}, {14, 5473}, {20, 98}, {30, 511}, {40, 13178}, {55, 13182}, {56, 13183}, {104, 10769}, {140, 6722}, {147, 3146}, {182, 2549}, {247, 1316}, {262, 11361}, {265, 15357}, {316, 9867}, {325, 13449}, {376, 671}, {381, 2482}, {382, 6033}, {485, 8997}, {486, 9739}, {487, 6230}, {488, 6231}, {546, 20399}, {547, 22247}, {549, 5461}, {550, 11623}, {575, 15048}, {576, 7737}, {618, 5479}, {619, 5478}, {631, 14061}, {641, 6251}, {642, 6250}, {944, 7983}, {946, 11711}, {962, 7970}, {1151, 8980}, {1152, 13967}, {1160, 22809}, {1161, 22810}, {1350, 11646}, {1351, 5477}, {1352, 10008}, {1385, 11725}, {1478, 10086}, {1479, 10089}, {1562, 17974}, {1569, 3095}, {1587, 19109}, {1588, 19108}, {1614, 3044}, {1657, 10991}, {1885, 12131}, {1916, 11257}, {2023, 13334}, {2080, 6781}, {2453, 5181}, {2936, 18534}, {3018, 5467}, {3023, 6284}, {3027, 7354}, {3058, 18969}, {3398, 7765}, {3455, 12083}, {3524, 9166}, {3529, 9862}, {3534, 11632}, {3543, 6054}, {3575, 5186}, {3627, 22505}, {3628, 15092}, {3830, 8724}, {3839, 23234}, {3845, 14160}, {4027, 6658}, {4297, 11599}, {4299, 10069}, {4302, 10053}, {4558, 8754}, {5026, 5480}, {5054, 14971}, {5055, 9167}, {5059, 5984}, {5066, 14162}, {5077, 19662}, {5085, 6034}, {5097, 18907}, {5149, 19130}, {5152, 11676}, {5182, 14853}, {5254, 13335}, {5355, 11842}, {5434, 12354}, {5469, 21157}, {5470, 21156}, {5471, 5615}, {5472, 5611}, {5476, 11159}, {5691, 9864}, {5864, 6777}, {5865, 6778}, {5870, 6320}, {5871, 6319}, {5976, 6248}, {5985, 15680}, {5986, 20062}, {5987, 20063}, {6228, 6229}, {6249, 8290}, {6459, 19056}, {6460, 19055}, {6699, 15359}, {6723, 11007}, {6771, 6772}, {6774, 6775}, {6776, 10754}, {7472, 16188}, {7802, 9991}, {7833, 22712}, {8356, 15819}, {8596, 11177}, {8782, 9873}, {9115, 20426}, {9117, 20425}, {9732, 12602}, {9733, 12601}, {9757, 9892}, {9758, 9894}, {9775, 14360}, {9834, 13176}, {9835, 13177}, {9838, 13184}, {9839, 13185}, {10352, 10358}, {10724, 10768}, {10733, 11005}, {11001, 12243}, {11500, 13173}, {11602, 22890}, {11603, 22843}, {11606, 12122}, {12041, 15535}, {12113, 13179}, {12114, 13180}, {12115, 13189}, {12116, 13190}, {12121, 18332}, {12177, 14928}, {12184, 12943}, {12185, 12953}, {12217, 12218}, {12303, 12979}, {12304, 12978}, {12383, 15342}, {12902, 15545}, {12910, 19474}, {12911, 19473}, {12972, 12985}, {12973, 12984}, {13075, 18974}, {13076, 18975}, {13081, 18988}, {13082, 18989}, {14033, 14561}, {14120, 16760}, {14538, 23004}, {14539, 23005}, {14568, 21445}, {14830, 15681}, {15687, 22566}, {15980, 18860}, {16001, 22513}, {16002, 22512}, {16163, 16278}, {18800, 20423}, {21158, 22510}, {21159, 22511}, {22501, 22591}, {22502, 22592}
= circumnormal isogonal conjugate of X(10425)
= [ 1.6307399271801790, 0.8970660651915741, -1.3736949346004030 ]
P03 = Q1 ( X(101) ) = INFINITY POINT OF THE LINE X(4)X(101)
= 6*(6*R^2-SW)*S^4+(SA-SW)*(54*R^2*SA-9*SA*SW-SW^2)*S^2-2*SB*SC*SW^3 : : (barys)
= on lines: {3, 126}, {4, 111}, {5, 6719}, {20, 1296}, {30, 511}, {104, 10779}, {113, 9129}, {376, 10717}, {381, 9172}, {382, 11258}, {944, 10704}, {946, 11721}, {1614, 3048}, {3146, 20099}, {3325, 7354}, {6019, 6284}, {6776, 10765}, {11818, 15563}, {15560, 18420}
= [ 0.9566282016884292, 1.9494441623744130, -1.7911358978077150 ]
P04 = ISOGONAL CONJUGATE OF P02
= (SB+SC)*(4*S^4+(6*SB-5*SW)*SB*S^2-SC*SA*SW^2)*(4*S^4+(6*SC-5*SW)*SC*S^2-SW^2*SA*SB) : : (barys)
= on the circumcircle and these lines: {3, 10425}, {4, 14384}, {98, 3566}, {99, 3564}, {107, 460}, {112, 1692}, {511, 3565}, {512, 3563}, {691, 1351}, {805, 9737}, {2715, 3053}, {2855, 9744}, {14265, 22456}
= circumperp conjugate of X(10425)
= antipode of X(10425) in the circumcircle
= trilinear pole of the line {6, 6132}
= [ 6.8313217550899690, 12.4183820720726500, -8.1095946857228270 ]
P05 = ISOGONAL CONJUGATE OF P03
= (a^10-(b^2+5*c^2)*a^8-(b^2-2*c^2)*(7*b^2+2*c^2)*a^6+(3*b^6+4*c^6+11*b^2*c^2*(b^2-2*c^2))*a^4+(b^2-c^2)*(6*b^6+5*c^6-b^2*c^2*(18*b^2+7*c^2))*a^2-(b^4-c^4)*(2*b^6+c^6-b^2*c^2*(6*b^2+c^2)))*(a^10-(5*b^2+c^2)*a^8+(2*b^2-c^2)*(2*b^2+7*c^2)*a^6+(4*b^6+3*c^6-11*b^2*c^2*(2*b^2-c^2))*a^4-(b^2-c^2)*(5*b^6+6*c^6-b^2*c^2*(7*b^2+18*c^2))*a^2+(b^4-c^4)*(b^6+2*c^6-b^2*c^2*(b^2+6*c^2)))*a^2 : : (barys)
= on the circumcircle and these lines: {112, 12593}, {1296, 2393}, {1499, 2373}
= [ 13.6264462119091400, 6.6867484520437040, -7.2777519288500960 ]
------------------------------------------
2)
Locus={Linf} ∪ {circumcircle} ∪ {K003 (McCay cubic)}
If P lies on the circumcircle then the point of intersection of the given lines Q2(P) = Q1(P)
ETC pairs (P,Q2(P)): (1, 145), (4, 4)
P06 = Q2( X(3) ) = COMPLEMENT OF X(22261)
= (S^2+(8*R^2-SA-2*SW)*SA)*(4*S^2+(SB+SC)*(8*R^2-SA-3*SW)) : : (barys)
= on lines: {2, 22261}, {140, 5449}, {570, 5254}, {6640, 15454}
= complement of X(22261)
= complementary conjugate of X(5562)
= [ 7.6505607846602070, 4.8904689202111350, -3.2760728250049630 ]
César Lozada
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