and 
be isogonal conjugates with respect to 
.
and 
be their respective pedal triangles. Let![\[ X_1=P_2Q_3\cap P_3Q_2,\quad X_2=P_1Q_3\cap P_3Q_1,\quad X_3=P_1Q_2\cap P_2Q_1 \]](http://latex.artofproblemsolving.com/7/f/e/7fe3ec4f69ec074635b2202daad9371e3bb8c72d.png)
Prove that the points 
,
and 
lie on the line 
.
, such that 
Prove that :
equality cases : ?
Prove that
![\[\left(a^{2}+2\right)\left(b^{2}+2\right)\left(c^{2}+2\right) \geq 9\left(ab+bc+ca\right)\]](http://latex.artofproblemsolving.com/e/9/a/e9a57f0700d37fbb28f7d219cc7ac7e52b43acd9.png)
holds for all positive reals 
,
,
.
Y by
Denote by 
the circumcenter of 
. The circumcircle of 
and 
intersects the lines 
and 
at points 
respectively. Similarly define points 
. Prove that lines 
have a common point.
the circumcenter of 
. The circumcircle of 
and 
intersects the lines 
and 
at points 
respectively. Similarly define points 
. Prove that lines 
have a common point.
