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 
.
meets 
.
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
Consider 
and 
three collinear points of the plane such that 
is between 
and . Let 
be the circle of diameter 
and 
a line that passes through , which does not intersect and is not perpendicular to line 
. The points 
and 
are, respectively, the feet of the altitudes drawn from and on the line . From draw the two tangent lines to , where 
is the closest tangency point to . Prove that the quadrilateral 
is cyclic if and only if the lines 
and 
are perpendicular.
and 
three collinear points of the plane such that 
is between 
and . Let 
be the circle of diameter 
and 
a line that passes through , which does not intersect and is not perpendicular to line 
. The points 
and 
are, respectively, the feet of the altitudes drawn from and on the line . From draw the two tangent lines to , where 
is the closest tangency point to . Prove that the quadrilateral 
is cyclic if and only if the lines 
and 
are perpendicular.
