AoPS Academy
be an acute scalene triangle , 
be the orthogonal projection of 
on 
, 
and 
are the midpoints of 
and 
respectively. Let 
and 
are points on the minor arcs 
and 
of the circumcircle of triangle respectively such that 
. Show that the circumcircles of triangles 
and 
are tangent if and only if lies on 
.
, 
, 
, ... be an infinite sequence of real numbers satisfying the equation 
for all 
, where and are two different positive reals., , , ... be bounded?
be a triangle with incircle 
and circumcircle 
. Let be any point on . Let the tangents from to meet at and . Prove that 
is also tangent to .
? Can the one passing through I be any of the infinite possibilities? How do you construct the one with A?
such that:
satisfying 
.
,
,
.
on 
and 
.
,
,
.
meet 
,
.
passes through 
,
has radius 