prove that![$\frac{a}{b} + \sqrt {\frac{b}{c}} + \sqrt[3]{{\frac{c}{a}}} > \frac{5}{2}$](http://latex.artofproblemsolving.com/6/a/6/6a657cb647e279b0745e3189c3f9d54aa25bf976.png)
and orthocenter 
,
be an arbitrary point on the circumcircle of triangle 
and that line 
is not parallel to line 
is perpendicular to 
.
and 
be the points on the circumcircle of triangle 
is a diameter, and 
and 
are parallel. Let 
be the midpoint of 
.
and 
are perpendicular.
and 
are perpendicular.
with 
,
intersects the extension of 
at point 
.
is the midpoint of 
,
intersects the circumcircle of 
.
.
Y by
The 
-excircle and 
-excircle of triangle 
(
) touch the sides of triangle 
and 
at points 
and 
, respectively. Let the lines 
and 
intersect at point 
. It is known that the symmetric of the point wrt line 
lies on the circle circumscribed around the triangle . Prove that 
.
-excircle and 
-excircle of triangle 
(
) touch the sides of triangle 
and 
at points 
and 
, respectively. Let the lines 
and 
intersect at point 
. It is known that the symmetric of the point wrt line 
lies on the circle circumscribed around the triangle . Prove that 
.
