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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 
.
, 
and 
of the rhombus 
, the points 
, 
and 
are selected respectively, so that 
and 
. Prove that the lines 
and 
pass through one point.
and 
as they are alternate angles
is external angle bisector of 
in 
be a point on 
extended such that 
as they are alternate angles
is external angle bisector of 
in 
is a rhombus, is angle bisector of 
is internal angle bisector of 
in 
and 
are concurrent and the point of concurrence is the excenter opposite to in 
