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be the Fermat point of a 
. Prove that the Euler line of the triangles 
, 
, 
are concurrent and the point of concurrence is 
, the centroid of .
(i know 
well). i want to study so if you guys have some easy or normal problems please send me
?
chess board, except for the 
central square, there is a checker. In one move, a checker can jump over adjacent checker (diagonally), and this adjacent checker (that was jumped over) is removed. Is it possible to come up with a sequence of moves, so that only one checker is left on the board?
touches the sides 
and 
at 
and 
respectively. Let 
be the 
-excenter of 
Prove that 
is perpendicular to the line determined by the circumcenters of 
and 
is the radical axis of 
and 
, so it suffices to show that 
has equal power with respect to these two circles. Let 
; then 
, meaning that a spiral similarity at 
takes 
to 
and the tangency point of the -excircle on 
to its tangency point on 
, meaning that the distance from to the foot from to is equal to the distance between the feet from 
onto , which is . Thus, 
. 
be other intersection of 
and 
we have 
cyclic. Similary 
cyclic.
is radical axis of 
and 
.