HYACINTHOS 26363

[Le Viet An]:

 

Let ABC be a triangle and DaDbDc the pedal triangle of I (intouch triangle).

Denote: 

 

Ab,Ac = the reflections of Da in B, C, resp.
Similarly Bc, Ba,Ca,Cb.
The lines BaCa, CbAb, AcBc bound triangle A'B'C'

[A' = AbCb /\ AcBc, B' = BaCa /\ BcAc, C' = CaBa /\ CbAb]

Then the Euler lines of A'AbAc, B'BcBa, C'CaCb are concurrent.

Which point is the point of concurrence?

 

[Angel Montesdeoca]

The Euler lines of A'AbAc, B'BcBa, C'CaCb are concurrent at W= 4 X(10) - 3 X(11)

W = (4 a^3 (b+c)-a^2 (b^2+14 b c+c^2)-4 a (b^3-2 b^2 c-2 b c^2+c^3)+(b^2-c^2)^2 : ... : ....)

W is the reflection of X(i) in X(j), for these {i, j}: {11,1145}, {149,3036}, {1317,100}, {1320,3035}, {5183,4394}, {6154,5541}, {7972,9945}, {7993,13226}, {12653,1387}.

W lies on lines X(i)X(j) for these {i, j}: {1, 6174}, {8, 190}, {10, 11}, {12, 10129}, {40, 550}, {56, 100}, {65, 10427}, {80, 4668}, {149, 2551}, {214, 3635}, {519, 1155}, {1000, 4413}, {1018, 4534}, {1320, 3035}, {1376, 13279}, {1387, 3624}, {2183, 3943}, {2254, 6366}, {2800, 3962}, {2829, 6361}, {3434, 13272}, {3617, 10707}, {3649, 10956}, {3679, 4679}, {3885, 8256}, {3893, 11362}, {3922, 12736}, {4701, 12732}, {4746, 12572}, {5082, 10953}, {5087, 6735}, {5252, 5856}, {5433, 10912}, {5434, 12648}, {5660, 11531}, {7091, 12641}, {7173, 13463}, {7972, 9945}, {7993, 13226}, {11500, 12245}, {12247, 12249}.

 (6 - 9 - 13) - search numbers  of W: (4.46619037228263, -2.98139348497599, 3.64338749199116).


Angel Montesdeoca