AoPS Academy
be a triangle. The incircle of touches the sides 
and 
at the points 
and 
, respectively. Let 
be the point where the lines 
and 
meet, and let 
and 
be points such that the two quadrilaterals 
and 
are parallelogram.
.
be an acute triangle with incenter 
, circumcenter 
, and circumcircle 
. Let 
be the midpoint of 
. Ray 
meets 
at 
. Denote by 
and 
the circumcircles of 
and 
, respectively. Line 
meets at 
and , while line 
meets at 
and 
. Assume that lies inside 
and 
. at and and the tangents to at 
and . Given that 
, prove that these four lines are concurrent on .
.
and 
Prove that
and 
be two oppositely oriented concentric equilateral triangles. Prove that the lines 
, 
, and 
intersect in one point.
