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be a triangle (
). 
are the altitudes of the triangle. The bisector of 
intersects 
at 
. Let 
be a point such that 
and 
. Prove that 
passes through the midpoint of 
.
with 
and at most one of them being zero, show that![\[\frac{1}{\max(a^2,b)}+\frac{1}{\max(b^2,c)}+\frac{1}{\max(c^2,a)}\geq\frac{27}{4}\]](http://latex.artofproblemsolving.com/9/e/1/9e14a9ef2d1a0a98006bcf716fa33c3d9ea29bd5.png)
then it can never pass through 
.
, 
is the external angle bisector of 
, while 
the one of 
, hence 
is the 
excenter of 
, consequently 
is the external angle bisector 
, similarly 
is the external angle bisector of . Under the circumstances, 
, done.
is the contact triangle)
almost trivially implies 
.
& 
.
