)
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 with circumcircle 
and let l be a line which does not intersect . Let 
be the foot of the perpendicular from the centre of to l. The side-lines 
and 
intersect l, respectively, at the points 
and 
different from . Prove that the circumcircles of triangles 
and 
have a common point different from or are mutually tangent at .
be a triangle with circumcircle 
and let l be a line which does not intersect . Let 
be the foot of the perpendicular from the centre of to l. The side-lines 
and 
intersect l, respectively, at the points 
and 
different from . Prove that the circumcircles of triangles 
and 
have a common point different from or are mutually tangent at .
