touch sides 
at 
,
lie on the line through 
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 
and 
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
Given the acute triangle 
, inscribed in circle with center 
. We construct the altitude 
, (
), and the angle bisector 
, (
), which intersects the circle at point 
. Denote with 
the intersection of the line 
with the circle. Prove that the lines 
and 
intersect on the circle, circumscribed around the triangle .
, inscribed in circle with center 
. We construct the altitude 
, (
), and the angle bisector 
, (
), which intersects the circle at point 
. Denote with 
the intersection of the line 
with the circle. Prove that the lines 
and 
intersect on the circle, circumscribed around the triangle .
Since 
this implies that 
is cyclic. Thus, noting that 
are well-known to be isogonal in 
we find that 
implies that 
are collinear as desired. 
is the center of the spiral similarity sending 
to 
