HYACINTHOS 29192

[Antreas P. Hatzipolakis]:

 

Let (O1) be the circle that passes through the vertices B and C,  and is tangent to the excircle (Ia). Define (O2) and (O3) cyclically.  Then the centroid of O1O2O3 is  ???

[Angel Montesdeoa]:


*** The centroid of O1O2O3 is  

W = X(3)X(9) ∩ X(72)X(15717)  

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

= (8R-r) X(3) + (4R+r) X(9) 

= lies on these lines: {3,9}, {72,15717}, {140,5806}, {165,16417}, {376,10157}, {517,549}, {631,31793}, {912,12100}, {942,3523}, {1071,10299}, {2800,31787}, {3524,11227}, {3530,9940}, {3579,22753}, {3628,31822}, {3646,24644}, {3940,10857}, {5447,31816}, {5927,10304}, {6924,31663}, {6961,31786}, {9729,31819}, {10167,15692}, {10176,10178}, {10202,15693}, {10855,16371}, {11220,15705}, {15712,31837}, {16192,25917}.

= midpoint of X(i) and X(j), for these {i, j}: {165, 5049}, {551, 10178}, {3576, 11227}

= reflection of X(10156) in X(549)
 
 (6 - 9 - 13) - search numbers of W: (5.65125675712842, 4.60456571858002, -2.15538413424567).
 
 Angel Montesdeoca