,
be all positive integers smaller than 
such that 
for all 
is the largest positive integer that divides both 
and 
.
.
,
and 
are the midpoints of sides 
and 
respectively. Point 
lies inside the triangle. Let 
.
.
,
,
of a hexagon 
inscribed in a circle 
intersect at a point 
,
.
be the radius of the circle tangent to segments 
and 
and internally tangent to 
,
,
,
,
are defined similarly. Prove that![\[
r_{AB}r_{CD} + r_{CD}r_{EF} + r_{EF}r_{AB} = r_{BC}r_{DE} + r_{DE}r_{FA} + r_{FA}r_{BC}.
\]](http://latex.artofproblemsolving.com/0/d/6/0d66d6685bbc97c7553e9ed3ba14ea0dc3328b49.png)
be the circumcentre, 
be the orthocentre, and 
be the centroid.
satisfy the following condition:
is constant.
Y by Adventure10, Mango247
Let 
be a sequnce of concentric circles such that the sequence 
, where 
denotes the radius of 
, is increasing and 
as 
. Let 
be a triangle inscribed in 
. extend the rays 
to meet 
in 
and 
respectively and form the triangle 
. Continue this process. Show that the sequence of triangles 
tends to an equilateral triangle as 
be a sequnce of concentric circles such that the sequence 
, where 
denotes the radius of 
, is increasing and 
as 
. Let 
be a triangle inscribed in 
. extend the rays 
to meet 
in 
and 
respectively and form the triangle 
. Continue this process. Show that the sequence of triangles 
tends to an equilateral triangle as 
