are lengths of sides of some triangle. Prove the inequality
where 
.
is the circumcenter of triangle 
is the projection of point 
onto line 
.
,
are 
,
,
,
intersects 
and 
at points 
and 
,
.
-
,
,
,
according to the following rule: If 
,
,
if 
(taking 
)
is the symbol other than 
if 
.
for which there exists some integer 
such that 
.
.
,
and 
be the feet of the altitudes of triangle 
,
respectively. Let 
and 
be the midpoints of 
and 
,
be the intersection of lines 
and 
.
.
Y by Adventure10, Mango247
Let 
be half-perimeter of triangle 
. Let 
and 
be a point's on ray's 
and 
, for which 
. Let 
is point, symmetric of point 
by circumcenter of . Prove, that perpendicular from to 
passes through incenter of .
Solution for problem here
be half-perimeter of triangle 
. Let 
and 
be a point's on ray's 
and 
, for which 
. Let 
is point, symmetric of point 
by circumcenter of . Prove, that perpendicular from to 
passes through incenter of .Solution for problem here
be the excenters of 
against 
Circumcircle 
and incenter 
of 
If 
denote the tangency points of the excircles 
with 
then 
is the circumcenter of 
is the Euler line of 
thus 
is a parallelogram with center 
i.e. 
Since 
it follows that circumcenter 
of 
is the midpoint of 
But, it's clear that 
are the tangency points of 
are antiparallel WRT 
since 
is an isosceles trapezoid with legs 
Thus, 
and together with 
it follows that 
