and 
Prove that
Prove that![$$\frac{a}{b}+ \frac{kb^2}{c^2} + \frac{c}{a}\geq 5\sqrt[5]{\frac{k}{16}}$$](http://latex.artofproblemsolving.com/8/b/e/8bed6605f3988f59b8395b091b92a8b9e8b40419.png)
Where 
and 
and 
,
be the intersection of lines 
and 
.
and 
and 
,
and 
are concurrent at a point on line 
,
and 
are the incenter and circumcenter of triangle 
Y by mathematicsy
In 
the angles 
and 
are acute. Let 
be the midpoint of 
. Point 
is on the half-line 
such that 
and 
. Perpendicular from 
to line 
intersects the line bisector of in 
. Prove that 
and 
are perpendicular.
the angles 
and 
are acute. Let 
be the midpoint of 
. Point 
is on the half-line 
such that 
and 
. Perpendicular from 
to line 
intersects the line bisector of in 
. Prove that 
and 
are perpendicular.
is cyclic, and thus, by the two right angles, 
is also cyclic. The result immediately follows.
