touch sides 
at 
,
lie on the line through 
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 
and 
respectively. 
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 Mango247
Consider acute triangles 
and 
, with 
, such that 
and 
are on opposite sides of line 
. Denote by 
the foot of the perpendicular line to 
through 
and by 
the foot of the perpendicular line to 
through 
. Let 
be the orthocenter of triangle and 
be the orthocenter of . Show that lines 
and 
are concurrent.
and 
, with 
, such that 
and 
are on opposite sides of line 
. Denote by 
the foot of the perpendicular line to 
through 
and by 
the foot of the perpendicular line to 
through 
. Let 
be the orthocenter of triangle and 
be the orthocenter of . Show that lines 
and 
are concurrent.
