[Antreas P. Hatzipolakis]:
Let ABC be a triangle, P a point and A'B'C' the pedal triangle of P.
Denote:
L1, L2, L3 = the Euler lines of AB'C', BC'A', CA'B', resp.
M1, M2, M3 = the parallels to L1, L2, L3 through P, resp.
N1, N2, N3 = the reflections of M1, M2, M3 in BC, CA, AB, resp.
Which is the locus of P such that N1, N2, N3 are concurrent?
H lies on the locus.
[César Lozada]:
Which is the locus of P such that N1, N2, N3 are concurrent at Z(P)?
Locus={a circumquartic through ETC’S 3,4,54,1147,2574,2575, the last two on the infinity }
Z(O) = X(110); Z(H) = X(186); Z(X(54)) = X(54)
Z(X(1147)) = (6*R^2-SA-SW)*SA*a : : (trilinears)
= cos(A)*(2*cos(A)*cos(B-C)-1) : : (trilinears)
= complementary conjugate of X(131)
= isogonal conjugate of X(1300)
= On the line at infinity, Gibert´s K039, K114, K339, Q097 and these lines:
(1,6238), (2,5654), (3,49), (4,52), (5,389), (6,4550), (20,6193), (26,6759),
(30,511), (40,6237), (51,381), (55,500), (56,1069), (69,4846), (74,323), (110,186),
(113,403), (125,1568), (131,1516), (140,9729), (143,546), (146,7731), (156,1658),
(161,1498), (182,7514), (232,1625), (265,1531), (373,5055), (376,2979), (378,1993),
(382,6243), (399,1495), (547,6688), (549,3819), (550,6101), (569,7503), (576,8548),
(578,7526), (944,9933), (974,6699), (1151,8909), (1350,8717), (1351,1597),
(1352,7706), (1478,20019), (1614,7488), (1994,7527), (3091,3567),
(3153,3448), (3193,7414), (3269,3289), (3357,9938), (3426,6391), (3519,3521),
(3523,7999), (3524,7998), (3545,5640), (3574,5576), (3818,9969), (3832,9781),
(4549,6776), (5054,5650), (5167,6033), (5609,7575), (5691,9896), (5752,6985),
(5870,9930), (5871,9929), (5921,6403), (5986,5999), (6030,7512), (6153,6288),
(6285,9931), (6407,8912), (6642,9786), (6644,9306), (8549,9926), (9873,9923
César Lozada
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