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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 
.
, so 
is the same length as the altitude from 
to 
, so 
. Similarly 
, so 
. Now we let be on the extension of such that 
. Then 
. Thus we have 
so 
.
are isosceles, 
and 
. If is midpoint of 
, then 
(angle - slope case), i.e. 
and marked angles are equal, so 
, thus 
.
such that 
.
is the incenter of 
and 
is the meet point of 
and the circumcircle of 
be on 
,
are perpendicular to 
,
.
as well.