HYACINTHOS 27538

[Tran Quang Hung]:

Let ABC be a triangle. and P a point.

A',B',C' are midpoints of PA,PB,PC, resp.

Q is the radical center of the circles (A',BC), (B',CA), (C',AB).

Then the line PQ passes through a fixed point on the Euler line.

Which is this point ?


[Angel Montesdeoca]:

**** The line PQ passes through a fixed point on Euler line:  

W = 9 a^4-4 (b^2-c^2)^2-5 a^2 (b^2+c^2) ::

 = 5 O - 4 H

= lies on these lines: {2,3}, {35,9656}, {36,9671}, {485,6445}, {486,6446}, {516,8148}, {999,4325}, {1154,12279}, {1159,1770}, {1384,7748}, {1482,9589}, {1539,15040}, {1975,7850}, {2777,12308}, {2794,14692}, {3068,9691}, {3070,9681}, {3295,4330}, {3411,5339}, {3412,5340}, {3527,13623}, {4296,9642}, {4299,9668}, {4301,10247}, {4302,9655}, {4309,6767}, {4316,12953}, {4317,6284}, {4324,12943}, {4746,11362}, {4816,5881}, {4902,7967}, {5024,7747}, {5093,12007}, {5840,8158}, {5925,13093}, {6101,12290}, {6241,13391}, {6243,10575}, {6361,12645}, {6407,13665}, {6408,13785}, {6417,6560}, {6418,6561}, {6455,6564}, {6456,6565}, {6472,9693}, {6496,10576}, {6497,10577}, {6519,8960}, {7737,9607}, {7756,9605}, {8717,11424}, {8976,9680}, {9606,15484}, {9654,15338}, {9669,15326}, {9690,13903}, {9692,13886}, {9703,13346}, {10113,15057}, {10263,15072}, {10574,13321}, {10627,15305}, {10706,15039}, {10733,15041}, {11439,15067}, {11455,11591}, {11485,16965}, {11486,16964}, {11820,12316}, {12002,16226}, {12121,15063}, {12162,13340}, {12902,16003}, {13348,16194}, {13598,14855}, {13881,15655}, {15046,15051}.

= the midpoint of X(3529) and X(5059).

= the reflection of X(i) in X(j), for these {i, j}: {3,1657}, {4,15704}, {381,11001}, {382,20}, {1657,3529}, {3146,550}, {3534,15683}, {3830,15681}, {5073,3}, {6243,10575}, {12290,6101}, {12645,6361}, {13093,5925}, {15640,549}, {15681,15685}, {15682,15686}, {15684,3534}.

 (6 - 9 - 13) - search numbers  of W:  (56.6182921146879, 55.5994117075985, -60.9827553685936)
 
 Angel Montesdeoca