)
are the altitudes of the triangle. The bisector of 
intersects 
at 
.
be a point such that 
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 
the altitude, 
the feet of the perpendiculars from 
to 
respectively. Let 
be a variable point on the line 
. The line through perpendicular to 
meets the lines at 
respectively.
i) Prove that circumcircle of
always passes the fixed point .
ii) Let
be another position of with corresponding points 
. Prove that the ratio 
is constant.
iii) The point
is symmetric to with respect to . The line through perpendicular to the line meets the line 
at 
. Prove that
.
be a triangle with 
the altitude, 
the feet of the perpendiculars from 
to 
respectively. Let 
be a variable point on the line 
. The line through perpendicular to 
meets the lines at 
respectively.i) Prove that circumcircle of
always passes the fixed point .ii) Let
be another position of with corresponding points 
. Prove that the ratio 
is constant.iii) The point
is symmetric to with respect to . The line through perpendicular to the line meets the line 
at 
. Prove that
.
and 
are similar, because 
and 
perpendicular to 
. So we have that 
.
decreases with the same value 
increases. Hence the ratio is constant.
