AoPS Academy
be a triangle with incircle 
, which touches sides 
and 
at points 
and 
, respectively. Choose points 
and 
on the line 
such that 
and 
. Let 
be the intersection of lines 
and 
. Define 
and 
as the intersections of 
and 
with lines 
and 
, respectively. Assume that 
is the intersection of 
and 
. Prove that 
.
be a function from the Euclidean plane to the real numbers such that 
for any acute triangle with circumcenter 
, centroid 
and orthocenter 
. Prove that 
is constant.
such that![\[
f(f(x)+y) = x+f(f(y))
\]](http://latex.artofproblemsolving.com/9/d/7/9d780faa5901646f7035e88400a9e3d8023f04ff.png)
for all 
.
be points in space which are not on a same plane and no any 3 points are not colinear.
is minimized at 
which is different from 
Prove that 
be points in space which are not on a same plane and no any 3 points are not colinear. is minimized at which is different from Prove that 
be points in space which are not on a same plane and no any 3 points are not colinear.
is minimized at 
which is different from 
Prove that 
and 
have coordinates 
and 
,where 
is the unit vector
and so on
for 
,which proves the claim.
such that 
.
