on the plane such that 
is an equilateral . Let 
intersect 
and 
at 
and 
respectively and 
intersect 
and 
at 
and 
respectively .
Prove that: 
is the sum of 
square numbers
Let 
and 
is the sum of 
is the sum of 
real number on the board such that each number is more than 
.
on the board and write 
.
Y by
Let 
be the feet of the altitudes of the triangle 
from 
, respectively. 
is the point on the circumcircle of the triangle , 
is the orthocenter of the triangle , and the incircle of triangle 
has radius 
. Let 
be the point such that and are on the opposite side of 
, line 
is perpendicular to the line , and the distance from to line is . Define 
and 
similarly. Prove that 
are collinear.
be the feet of the altitudes of the triangle 
from 
, respectively. 
is the point on the circumcircle of the triangle , 
is the orthocenter of the triangle , and the incircle of triangle 
has radius 
. Let 
be the point such that and are on the opposite side of 
, line 
is perpendicular to the line , and the distance from to line is . Define 
and 
similarly. Prove that 
are collinear.
midpoint of 
.
are on nine-point circle of 
and 
.
be midpoint of 
. so it sufficient to prove that 
be point that incircle of 
respectively.
be projection 
respectively.
