AoPS Academy
opposite the vertex 
be tangent to the side 
at the point 
. Define the points 
on 
and 
on 
analogously, using the excircles opposite 
and 
, respectively. Suppose that the circumcentre of triangle 
lies on the circumcircle of triangle . Prove that triangle is right-angled.
be the circumcircle of triangle , 
is the midpoint of and 
be the second intersection of 
and . Tangent line of at intersects at 
, let 
be a transversal of and 
be intersections of 
with . Let 
be a tangent line of . Prove that 
is cyclic
be positive integers such that 
and 
. Find the maximum possible value of 
.
with area 
and a point 
in the same plane, determine the area of polygon 
where 
is the image of 
around 
by 
