[APH]:
Let ABC be a triangle, P a point and A'B'C' the pedal triangle of P.
Denote:
Ba, Ca = the orthogonal projections of B', C' on AB, AC, resp.
Which is the locus of P such that the Euler line La of ABaCa is perpendicular
to BC?
N lies on the locus.
We have three loci corresponding to A,B,C
Which points, other than N, are their intersections?
[César Lozada]:
For A,B,C the required loci are the correspondent cevian of N, together with the line at infinity.
For P=N the Euler lines La, Lb, Lc concur at:
Z= (cos(2*A)-2)*cos(B-C)+cos(A) :: (trilinears)
= X(4)+3*X(51)
= midpoint of X(i),X(j) for these {i,j}: {4,389}, {5,5446}, {52,5907}, {143,546}, {1112,7687}, {5480,9969}
= reflection of X(i) in X(j) for these (i,j): (5447,3628), (9729,5462)
= On lines:
(3,5943), (4,51), (5,141), (6,1598), (20,5640), (25,578), (30,5462), (49,7545),
(52,381), (54,1495), (64,3531), (68,3818), (140,6688), (143,546), (155,576),
(181,3073), (182,7387), (373,631), (382,9730), (403,3574), (428,6146), (517,5795),
(550,5892), (568,3843), (569,7517), (575,7530), (970,3560), (973,1112), (1092,1995),
(1154,3850), (1173,1199), (1181,5198), (1597,3357), (1656,3819), (1843,3089),
(1864,1871), (1872,2262), (2818,7686), (2979,5056), (3060,3091), (3072,3271),
(3090,3917), (3098,7393), (3627,5946), (3628,5447), (3832,5889), (3845,6102),
(3851,5891), (3858,5876), (3861,5663), (5067,5650), (5071,7999), (5752,6913),
(6403,6622), (6530,6750), (6995,9833), (7486,7998), (7529,9306)
= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k):
(4,51,389), (4,1093,8887), (4,3567,185), (4,9781,51), (6,1598,6759), (51,185,3567),
(52,381,5907), (185,3567,389), (1597,9786,3357), (1598,3527,6), (3060,3091,5562),
(3851,6243,5891), (5198,9777,1181)
= [ -1.328685665701012, -1.74991120893009, 5.465381010721065 ]
César Lozada
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