HYACINTHOS 25063

Let ABC be a triangle and A'B'C' the pedal triangle of a point P. Denote: A", B", C" = the orthogonal projections of A, B, C on PA', PB', PC', resp. Ab, Ac = the orthogonal projections of A" on AC, AB, resp. Bc, Ba = the orthogonal projections of B" on BA, BC, resp. Ca, Cb = the orthogonal projections of C" on CB, CA, resp. Which is the locus of P such that the lines AbAc, BcBa, CaCb are concurrent? The entire plane? (The parallels to AbAc, BcBa, CaCb through A, B, C, resp, concur at O.) [César Lozada]: > Which is the locus of P such that the lines AbAc, BcBa, CaCb are concurrent? The entire plane? Yes. For P=u:v:w (trilinears) they concur at Z(P) = a*(b^2+c^2-a^2)*((b^2+c^2)*a* u+b^3*v+c^3*w) : : ETC-pairs (P,Z(P)): (2,3917), (3,1216), (4,3), (5,5447), (8,72), (20,5562), (51,10691), (64,5907), (66,141), (69,69), (76,7767), (149,3937), (150,1565), (193,6467), (194,4173), (265,6699), (315,3933), (316,6390), (497,3784), (511,3564), (512,525), (513,521), (516,916), (517,912), (520,8057), (523,520), (524,8681), (525,8673), (526,9033), (674,9028), (690,9517), (693,4131), (850,3265), (900,8677), (924,523), (962,1071), (1154,539), (1370,394), (1503,511), (1510,6368), (1853,3819), (1899,1368), (2390,519), (2393,524), (2550,3781), (2574,2574), (2575,2575), (2777,5663), (2778,2771), (2781,542), (2818,952), (2892,5181), (3146,185), (3267,4143), (3421,3940), (3434,63), (3436,78), (3448,125), (3566,512), (3827,518), (3852,732), (5080,5440), (5082,3927), (5189,3292), (5207,6393), (5596,3313), (5878,550), (5889,6146), (6000,30), (6001,517), (6225,20), (6243,10116), (6327,306), (6515,1899), (6759,10627), (6776,9967), (7391,184), (8674,2850), (9001,9051), (9002,9031), (9812,10167), (9833,6101), (10733,974), (11206,2979), (11411,68), (11433,7386), (11442,343), (11550,6676) Others (trilinears): Z(I) = reflection of X(389) in X(9940) = a*(-a^2+b^2+c^2)*((b^2+c^2)*a+ b^3+c^3) : : = On lines: {1,7186}, {3,73}, {10,8679}, {36,1408}, {51,5439}, {52,10202}, {57,5752}, {72,3917}, {185,10167}, {389,9940}, {511,942}, {517,4292}, {674,3874}, {912,1216}, {971,5907}, {1046,3792}, {1469,5711}, {2390,3878}, {2979,3868}, {3781,3927}, {3819,5044}, {3876,7998}, {3888,5015}, {3916,3937}, {9729,11227} = [ 4.873132226152876, 4.07475605652914, -1.429458430837121 ] Z(X(6)) = reflection of X(389) in X(182) = ((b^2+c^2)*a^2+b^4+c^4)*a*(-a^ 2+b^2+c^2) : : = SW*X(3)+(4*R^2-SW)*X(6) = On lines: {2,1843}, {3,6}, {5,3867}, {22,1974}, {51,3618}, {69,305}, {110,1205}, {141,1368}, {159,9306}, {193,2979}, {232,7467}, {524,6665}, {631,6403}, {1038,1469}, {1040,3056}, {1216,3564}, {1352,6643}, {1353,6101}, {1503,5907}, {2386,7761}, {2854,3631}, {3547,10110}, {3589,5943}, {3619,5650}, {3620,7998}, {3630,9027}, {3763,9973}, {3779,10319}, {3781,5227}, {3784,7289}, {5020,7716}, {5140,7841}, {5159,8705}, {5480,6823}, {5562,6776}, {5596,6000}, {5800,10441}, {5921,11444}, {6688,9971}, {7484,9813}, {7485,8541} = midpoint of X(i) and X(j) for these {i,j}: {3,9967}, {6,3313}, {69,6467}, {110,1205}, {1351,10625}, {1353,6101}, {5562,6776} = reflection of X(i) in X(j) for these (i,j): (389,182), (1843,9822), (9969,3589), (9971,6688) = anticomplement of X(9822) = complement of X(1843) = 2nd Brocard circle-inverse-of-X(9917) = {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (2,1843,9822), (1350,5085,1192), (3589,9969,5943), (3917,6467,69) = [ 4.756023309737503, 4.35297181870473, -1.568019073997740 ] César Lozada