HYACINTHOS 26119

[Le Viet An]:

Let ABC be a triangle and IaIbIc the antipedal triangle of I (excentral triangle).

    Denote:

    Ao, Bo, Co = the orthogonal projections of Ia, Ib, Ic on BC, CA, AB, resp.

    A1, B1, C1 = the circumcenters of IBC, ICA, IAB, resp.

    A2, B2, C2 = the reflections of A1, B1, C1 in A0, B0, C0, resp.

    H, A2, B2, C2 are concyclic.

    Which point is the center of the circle?

 

[Angel Montesdeoca]:

***  The center of the circle ( H, A2, B2, C2) is:
 
W = (a (a^6+a^5 (b+c)-a^4 (4 b^2+b c+4 c^2)-2 a^3 (b-c)^2 (b+c)+a^2 (5 b^4+b^3 c-2 b^2 c^2+b c^3+5 c^4)+a (b-c)^4 (b+c)-2 (b^2-c^2)^2 (b^2+c^2)):...:...),

W lies on lines X(i)X(j) for these {i,j}:  {3, 191}, {8, 30}, {21, 10246}, {40, 12786}, {56, 1749}, {79, 10895}, {153, 5690}, {355, 12745}, {499, 3649}, {517, 7701}, {758, 1482}, {1046, 5492}, {1656, 11263}, {2095, 5789}, {2475, 5790}, {3065, 5697}, {3467, 5902}, {3577, 6597}, {3579, 4005}, {3678, 12515}, {3811, 11849}, {3878, 12773}, {4880, 9955}, {5055, 5221}, {12331, 12342}.

with (6-9-13)-search number (2.54568752814939, -0.0785059815758744, 2.52008207154488).

W is the reflection of X(3) in X(191),

X(191) = isogonal-conjugate-with- respect-to-excentral-triangle of X(3) (Randy Hutson, 9/23/2011)

Angel Montesdeoca