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)
satisfying 
is inscribed in circle 
.
denote its incenter and 
its 
-
denote the midpoint of arc 
.
meets 
.
at 
and 
at 
and 
respectively. The circumcircle of 
meets 
a second time at 
,
meets 
a second time at 
.
passes through the antipodal to 
that satisfy 
with 
,
and area 
.
with diameter 
at 
,
and extend to meet 
and 
is the diameter of 
of 
,
of 
and 
Prove that
Let 
Prove that
Let 
Prove that
Let 
Prove that
.
lie on 
respectively, such that 
.
to meet the circumcircle of 
respectively. Let the circumcircles of 
meet at points 
.
to 
,
,![$[ABC]$](http://latex.artofproblemsolving.com/d/3/3/d33cc80fa8f093e155c5be46d2e5d9da3d7e1ef5.png)
.
with 
,
to meet the circumcircle of 
at 
,
.
Y by Adventure10
Suppose that 
is a cyclic quadrilateral and 
. Points 
and 
belong to the segments 
and 
respectively, and 
. Segments 
and 
are height and median of triangle 
, respectively. 
is the point symmetric to 
with respect to 
. Prove that the lines and 
are parallel.
is a cyclic quadrilateral and 
. Points 
and 
belong to the segments 
and 
respectively, and 
. Segments 
and 
are height and median of triangle 
, respectively. 
is the point symmetric to 
with respect to 
. Prove that the lines and 
are parallel.
is the bisector of angle 
and 
therefore 
is the B-excenter of triangle 
.
of triangle 
.
then it easy to show that 
.
,
is cyclic, 
is a bisector of 
.
,
,
,
,
-
.
and 
are altitudes of 
and 
is cyclic 
.
is cyclic 
.
and 
lie on the same line and 
is an altitude.
such that 
,
and 
are the projections of 
and 
on 
,
.
and 
and 
.
.
is cyclic. We know that 
is cyclic, so
.
is the excenter of triangle 
