HYACINTHOS 24817

[Antreas P. Hatzipolakis]:

   
    Let ABC be a triangle, P a point and A'B'C' the pedal triangle of P.

    Which is the locus of P such that the reflections of AA', BB', CC' in PA', PB', PC', resp. are concurrent?

    N lies on the locus. The point of concurrence is the G.
   
   

[Angel Montesdeoca]:

**** The locus of P such that the reflections of AA', BB', CC' in PA', PB', PC', resp. are concurrent (at Q) is K127, the locus of point P such that the reflection triangle of ABC in the sidelines of ABC and the pedal triangle of P are perspective.
   
Points on the K127:   X(4), X(5), X(3146)    
   
If P=X(4), Q=X(4)
If P=X(5), Q=X(2)
If P=X(3146),  Q=(7 a^2-3 (b^2+c^2) : ... : ...)
Q = midpoint of X(382) and X(5073)
Q = reflection of X(i) in X(j) for these {i,j}: {2,3543}, {3,3627}, {4,382}, {8,5691}, {20,4}, {145,962}, {376,3830}, {550,3853}, {1657,5}, {3529,3}, {5059,20}, {5925,6247}, 5984,148}, {6225,5895}, {6240,7553}, {6241,52}, {6361,355}, {9862,6321}, {9961,65}

Angel Montesdeoca