AoPS Academy
and 
of triangle 
intersect in 
. A perpendicular 
from to 
is drawn. Circumcircles and 
intersect at 
and 
.
.
be a scalene triangle. The incircle of 
is tangent to sides 
, 
, 
at 
, respectively. Let 
be a point on the incircle of such that 
. If lines 
and intersect at 
, prove that 
is parallel to 
.
that is inscribed in 
. The incircle 
is tangent to 
at 
respectively. A line through 
parallel to intersects at 
, and 
intersects again at 
. Let 
is the midpoint of . 
be the second intersection of 
with . 
intersects again at 
. Prove that the line passing through perpendicular to 
bisects 
.
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