ADGEOM 3941

[Tran Quang Hung]:

Let ABC be a triangle with circumcenter O and orthocenter H.

A',B',C' are inversion image of A,B,C through circle diameter OH.

Then circumcenter of triangle A'B'C' lies on OH. Which is this point ?

 

 

[Angel Montesdeoca]:

**** The circumcenter of triangle A'B'C' is

O' = (a^2 (a^8
                -2 a^6 (b^2+c^2)
               +5 a^4 b^2 c^2
               +a^2 (2 b^6-b^4 c^2-b^2 c^4+2 c^6)
               -(b^2-c^2)^2 (b^4+4 b^2 c^2+c^4)) : ... : ...).
 
 
O' is the reflection of X(13564) in X(3).
O' lies on lines X(i)X(j) for these {i, j}: {2, 3}, {36, 9628}, {49, 399}, {54, 5663}, {74, 13434}, {185, 567}, {265, 13403}, {511, 12307}, {568, 7689}, {569, 3357}, {575, 5621}, {1092, 4550}, {2777, 3521}, {2935, 9730}, {3167, 9938}, {3567, 11454}, {3581, 5446}, {5012, 13491}, {5609, 9705}, {6000, 10274}, {6102, 11440}, {7691, 13391}, {8254, 11805}, {9655, 9672}, {9659, 9668}, {9703, 11441}, {10540, 13367}, {10564, 11793}, {10628, 11560}, {10982, 13321}, {11439, 11464}, {12006, 12041}, {12038, 12302}.

 (6 - 9 - 13) - search numbers  of O': (-117.833758800421, -118.392430292339, 139.989466438336).
 
 Angel Montesdeoca