,
touch sides 
at 
,
perpendicular to 
.
be the intersections of 
with 
,
be the projections of 
onto line 
be the second intersections of 
with the incircle 
be the intersection of 
and 
.
bisects segment 
.
be the incentre of 
such that the incircle touches the sides 
meets 
and 
meet the incircle at 
respectively. 
and 
intersect at 
Prove that 
bisects 
then prove 
![$P\in\mathbb{Z}[x]$](http://latex.artofproblemsolving.com/f/3/d/f3d763f4197792a2956273485eb1e94700a43044.png)
has degree 
having 
.
is the leading coefficient of 
Y by
Two circle 
and 
intersect at points 
and 
, point 
is not on the line 
. Line 
intersects again and at points 
and 
respectively, line 
intersects again and at points 
and 
respectively and all the points mentioned so far are different. The centers of the circles circumscribed around the triangles 
and 
are 
and 
respectively. Prove that 
is perpendicular to .
and 
intersect at points 
and 
, point 
is not on the line 
. Line 
intersects again and at points 
and 
respectively, line 
intersects again and at points 
and 
respectively and all the points mentioned so far are different. The centers of the circles circumscribed around the triangles 
and 
are 
and 
respectively. Prove that 
is perpendicular to .
and 
are tangent.
