HYACINTHOS 29188

[Angel Montesdeoca]:

FromAoPS:

 Let (Oa) the circle that passes through vertices B and C,  and is tangent to the incircle. Define (Ob) and (Oc) cyclically.  Then the centroid of  OaObOc is  

 

W = X(1)X(3) ∩ X(140)X(9947)

 

= a (-a^5 (b+c)+a^4 (b^2-20 b c+c^2)+2 a^3 (b^3+2 b^2 c+2 b c^2+c^3)-2 a^2 (b^4-11 b^3 c+4 b^2 c^2-11 b c^3+c^4)-a (b-c)^2 (b^3+5 b^2 c+5 b c^2+c^3)+(b-c)^4 (b+c)^2) : : (barys)

=  (4R+r) I + (8R-r) O  

= lies on these lines: {1,3}, {140,9947}, {515,10156}, {551,10178}, {631,18908}, {971,10165}, {1125,15726}, {2801,5044}, {3555,15717}, {3646,12684}, {3817,3824}, {3848,28164}, {5265,7671}, {5886,21151}, {5918,25055}, {5927,16845}, {6916,18527}, {9858,30478}, {10864,16853}, {11220,17558}, {13369,31821}.

= the midpoint of X(i) and X(j), for these {i, j}: {165, 5049}, {551, 10178}, {3576, 11227}
 
 (6 - 9 - 13) - search numbers  of W: (4.98829624962058, 4.41354319946759, -1.71715600216422).
 
 Angel Montesdeoca