,
touch sides 
at 
,
perpendicular to 
be the intersections of 
with 
,
be the projections of 
onto line 
be the second intersections of 
with the incircle 
and 
.
bisects segment 
.
be the incentre of 
such that the incircle touches the sides 
and 
meet the incircle at 
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 circles 
and 
intersect at points 
and 
. Circle 
is tangent to 
at 
and at 
, and intersects at 
and 
. 
intersects 
and at 
and 
, respectively. 
is the point on diametrically opposite to . 
intersects at 
, and 
passes through at 
. Show that 
, 
, and 
are concurrent.
and 
intersect at points 
and 
. Circle 
is tangent to 
at 
and at 
, and intersects at 
and 
. 
intersects 
and at 
and 
, respectively. 
is the point on diametrically opposite to . 
intersects at 
, and 
passes through at 
. Show that 
, 
, and 
are concurrent.
.
.
we have 
are concurrent at 
.
and the concurrency immediately follows from the prism lemma. 
