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, let the incircle 
touch sides 
at 
, respectively. Let 
lie on the line through 
perpendicular to 
. Let 
be the intersections of 
with 
, respectively. Let 
be the projections of 
onto line . Let 
be the second intersections of 
with the incircle . Let 
be the intersection of 
and 
. Prove that the line 
bisects segment 
.
be the incentre of 
such that the incircle touches the sides at the points respectively. The perpendicular to at 
and 
meets at 
and 
respectively. 
and 
meet the incircle at and 
respectively. 
and 
intersect at 
Prove that 
bisects 
![$P\in\mathbb{Z}[x]$](http://latex.artofproblemsolving.com/f/3/d/f3d763f4197792a2956273485eb1e94700a43044.png)
has degree 
having real roots in 
. Suppose 
is the leading coefficient of . Show that, 
with face and edges 
. Denote by 
the diagonal intersection points of the quadrilaterals 
respectively. We need to show that 
intersect at a point.
and 
lie in the plane 
they intersect. Analogously intersect also the pairs 
and 
thus the lines intersect at a point (they share the point of intersection of the planes 
) because these three lines don't lie in a plane.
then prove 
