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 Adventure10, Mango247
In a triangle 
, let 
be a point on its circumscribed circle (on the arc 
that does not contain 
). Let 
and 
be the orthocenters of triangles 
and 
, respectively. Let 
and 
. If 
and 
are the midpoints of 
and 
, respectively, prove that 
and 
intersect at their midpoint.
, let 
be a point on its circumscribed circle (on the arc 
that does not contain 
). Let 
and 
be the orthocenters of triangles 
and 
, respectively. Let 
and 
. If 
and 
are the midpoints of 
and 
, respectively, prove that 
and 
intersect at their midpoint.
