AoPS Academy
with circumcenter 
and orthocenter 
, let 
be an arbitrary point on the circumcircle of triangle such that does not lie on line 
and that line 
is not parallel to line 
. Let 
be the point on the circumcircle of triangle such that 
is perpendicular to , and let 
be the point on line 
such that 
. Let 
and 
be the points on the circumcircle of triangle such that 
is a diameter, and 
and 
are parallel. Let 
be the midpoint of 
.
and 
are perpendicular.
and 
are perpendicular.
with 
, a tangent to the circumcircle of at point 
intersects the extension of 
at point 
. 
is the midpoint of 
, and 
intersects the circumcircle of at 
. Prove that 
.
and the altitude by 
We are asked to show that 
However, 
where 
Also, we know that 
Therefore, 
is equivalent to
by the statement of the problem we have 
and 
Thus, 
is equivalent to 
which can be rewritten as (1-x)(1-y)+(1-x)(1-y)(x^3+y^3+5xy-5x-5y+4)\ge 0.\ (3)$](http://latex.artofproblemsolving.com/c/9/2/c925228ce8276a896698f351889bd6457b4052d2.png)
because all other terms in 
are 
But for 
this is obvious because ![$f(x,y)=(x+y-1)^3+(1-x)(1-y)\left[5-3(x+y)\right]\ge 0,$](http://latex.artofproblemsolving.com/f/0/1/f0150f9a8e2d45f6f70752db8f625b66ff384b78.png)
while for 
we have 
holds under the conditions of the problem. Equality occurs when the triangle is an equilateral triangle.
prove that![$\frac{a}{b} + \sqrt {\frac{b}{c}} + \sqrt[3]{{\frac{c}{a}}} > \frac{5}{2}$](http://latex.artofproblemsolving.com/6/a/6/6a657cb647e279b0745e3189c3f9d54aa25bf976.png)
we have 
and 
Prove that the ratio of circumcircle radius to incircle diameter is at least as big as the length of the centroidal axis 
to the altitude 
When do we have equality?
