HYACINTHOS 26382

[Le Viet An]:

Let ABC be a triangle, MaMbMc the cevian triangle of G and Fe the Feuerbach point.
The line IG intersects MbMc, McMa, MaMb at Ka, Kb, Kc, resp.
The perpendiculars to IA, IB, IC from Ka, Kb, Kc, resp. bound a triangle A'B'C'.

Then
The circumcircle of A'B'C' touches the incircle at a point T.
1. Which point is the center of the circle?
2. Which point is T?

Let K be the orthocenter of A'B'C'.
The points I, T, K, Fe are concyclic
3. Which point is the center of the circle (lying on the line IG) ?


[Peter Moses]:


Hi Antreas,

A' = {2 a (2 a-b-c) (b-c)^2,-a^4+3 a^3 b-3 a^2 b^2-a b^3+a^3 c-3 a^2 b c+7 a b^2 c-b^3 c+a^2 c^2-3 a b c^2-a c^3+b c^3,-a^4+a^3 b+a^2 b^2-a b^3+3 a^3 c-3 a^2 b c-3 a b^2 c+b^3 c-3 a^2 c^2+7 a b c^2-a c^3-b c^3}.

1).
a^6 b-4 a^5 b^2-2 a^4 b^3+7 a^3 b^4-3 a b^6+b^7+a^6 c-2 a^5 b c+11 a^4 b^2 c-5 a^3 b^3 c-11 a^2 b^4 c+7 a b^5 c-b^6 c-4 a^5 c^2+11 a^4 b c^2-18 a^3 b^2 c^2+13 a^2 b^3 c^2+3 a b^4 c^2-3 b^5 c^2-2 a^4 c^3-5 a^3 b c^3+13 a^2 b^2 c^3-14 a b^3 c^3+3 b^4 c^3+7 a^3 c^4-11 a^2 b c^4+3 a b^2 c^4+3 b^3 c^4+7 a b c^5-3 b^2 c^5-3 a c^6-b c^6+c^7:: 

on lines {{3, 8}, {10, 3667}, {40, 2957}, {517, 1647}, {5697, 6018}}.

2).
(2 a-b-c)^2 (b-c)^2 (a+b-c) (a-b+c):: 

on lines {{8, 6079}, {11, 3667}, {55, 2743}, {56, 2222}, {57, 3322}, {214, 519}, {244, 1365}, {513, 1357}, {517, 6018}, {1155, 3021}, {1358, 3676}, {1362, 3660}, {1366, 1447}, {1397, 2720}, {1647, 3259}, {3328, 3675}, {4014, 5577}, {5061, 5211}, {5433, 6789}, {6790, 7288}}.
midpoint of X(2718) and X(6788).
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (244,6075,7336).
on the incircle.
isoconjugate of X(j) and X(j) for these (i,j): {{644, 4638}, {679, 6065}, {765, 1318}, {1320, 9268}, {2316, 5376}, {3257, 5548}, {3939, 4618}}.
crosspoint of X(i) and X(j) for these (i,j): {{3676, 3911}}.
crosssum of X(i) and X(j) for these (i,j): {{55, 5548}, {2316, 3939}}.
barycentric product X(i)X(j) for these {i,j}: {{279, 4542}, {1086, 1317}, {1358, 4370}, {1647, 3911}, {3676, 6544}}.
barycentric quotient X(i)/X(j) for these {i,j}: {{900, 4582}, {1015, 1318}, {1017, 6065}, {1317, 1016}, {1319, 5376}, {1357, 2226}, {1404, 9268}, {1647, 4997}, {1960, 5548}, {2087, 1320}, {3251, 644}, {3669, 4618}, {4370, 4076}, {4542, 346}, {4543, 6558}, {6544, 3699}}.
reflection of X(1357) in the OI line.
reflection of X(11) in the Nagel line

3).
(2 a-b-c) (a^3-a^2 b-3 a b^2-b^3-a^2 c+7 a b c+b^2 c-3 a c^2+b c^2-c^3):: 

on lines {{1, 2}, {11, 11717}, {900, 1387}}.
midpoint of X(1) and X(1647).
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,6788,3244).
 

Best regards,
Peter Moses.