be a function such that for every regular 
-
we have 
.
for all reals 
.
)
are the altitudes of the triangle. The bisector of 
intersects 
at 
.
and 
.
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)
Y by Adventure10, Mango247
Let 
be a triangle and 
be the angle bisector of 
, with 
on 
. Let 
be a point on segment such that 
. Through draw 
a parallel line to and let 
be a point in inside the triangle. Let 
be the point where 
intersects 
and 
be the point where 
intersects 
. Show 
.
be a triangle and 
be the angle bisector of 
, with 
on 
. Let 
be a point on segment such that 
. Through draw 
a parallel line to and let 
be a point in inside the triangle. Let 
be the point where 
intersects 
and 
be the point where 
intersects 
. Show 
.
,
be points on 
,
,
.
is parallel to angle bisector of 
at 
such that 
lands on 
.
denote the new positions of 
.
.
.
is a parallelogram. Thus, it is enough to prove that 
is parallel to the 
angle bisector. This is true since
,
and 
.
and Let 
,
and let 
as the Newton Line of the complete quadrilateral 
Hence we're done!!
