)
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 
. Altitudes 
meet at 
. 
cuts 
at 
. 
is the midpoint of , 
cuts 
in 
.
a) Prove that
.
b) Let
be the circumcircle of triangle , 
be the circumcircle of triangle 
. 
cuts circle at 
, 
cuts at 
, 
cuts circle of diameter at 
. Prove that 
lie on the same circle.
be a triangle with 
. Altitudes 
meet at 
. 
cuts 
at 
. 
is the midpoint of , 
cuts 
in 
.a) Prove that
.b) Let
be the circumcircle of triangle , 
be the circumcircle of triangle 
. 
cuts circle at 
, 
cuts at 
, 
cuts circle of diameter at 
. Prove that 
lie on the same circle.
= 
= 
,
,
,
,
lie on a circle
or 
= 
+ 
= 
+ 
= 
+ 
= 
= 2 
)
= 
= 
= 
. 
tangents 
)
= 
= 
= 
= 
= 
. 
then: 
= 
= 
= 
= 
. 
= 2 
= 
= 
