HYACINTHOS 23783

 

[APH]

Pedal triangle version (2^nd):

 

Let ABC be a triangle and A'B'C' the pedal triangle of I.

Denote:

Ha, Hb, Hc = the orthocenters of AB'C', BC'A', CA'B', resp.

The Euler lines of AHbHc, BHcHa, CHaHb and  A'B'C' [= OI line of ABC]

are concurrent.

 

 

[Seiichi Kirikami]:
 

1^st barycentric coordinate of 2^nd case =

a (a^5 (b + c) - (b - c)^4 (b + c)^2 - a^4 (b^2 + c^2) - 2 a^3 (b^3 + c^3) + 2 a^2 (b^4 - b^3 c - 2 b^2 c^2 - b c^3 + c^4) +a (b^5 - b^4 c - b c^4 + c^5)).

Search number of triangle {6,9,13} = 1.508612666646547..

 

Best regards, Seiichi.

 

P. S. 1^st case is difficult to solve because its expressions can not exclude triangular functions.

[Peter Moses]:

on lines {{2,3},{113,1209},{127,3934},{131,137},{184,9927},{185,5449},{216,1879},{265,6146},{1060,7741},{1062,7951},{1216,1568},{3574,5446},{3580,6102},{5448,5562},{5476,8538},{5893,7728}}

 

Best regards,

Peter Moses