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 Adventure10
Let 
be a trapezoid with the longer base 
such that its diagonals 
and 
are perpendicular. Let 
be the circumcenter of the triangle 
and 
be the intersection of the lines 
and 
. Prove that 
.
be a trapezoid with the longer base 
such that its diagonals 
and 
are perpendicular. Let 
be the circumcenter of the triangle 
and 
be the intersection of the lines 
and 
. Prove that 
.This post has been edited 1 time. Last edited by parmenides51, Dec 23, 2022, 9:01 PM
is tangent to 
at point 
,
is tangent to 
.
