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 
meets 
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 Adventure10, Mango247
Let 
be the midpoint of the base 
of the isosceles acute triangle 
. Choose point 
on segment , and let 
be the circumcenter of triangle 
. Prove that the line through perpendicular to 
, the line through perpendicular to 
, and the line through 
parallel to 
are concurrent.
be the midpoint of the base 
of the isosceles acute triangle 
. Choose point 
on segment , and let 
be the circumcenter of triangle 
. Prove that the line through perpendicular to 
, the line through perpendicular to 
, and the line through 
parallel to 
are concurrent.
at 
it is trivial to verify 
so 
and 
are cyclic.
intersect the line through 
By radical axis it suffices to show 
lies on 
and 
thus 
is the reflection of 
and 
is the reflection of 
thus 
and it suffices to show 
at 
Since 
and 
are both perpendicular to 
they are parallel. 
so 
lies on 
Notice that 
is right and it easily follows that 
is the perpendicular bisector of 
implying the result. 
