be a cyclic quadrilateral, and let 
,
,
,
be the midpoints of 
,
,
respectively. Let 
,
,
and 
be the orthocenters of triangles 
,
,
and 
,
have the same area.
with real coefficients is the average of two monic polynomials of degree 
of positive integers, for which![$$(a, b) + 3[a, b] = a^3 - b^3$$](http://latex.artofproblemsolving.com/b/8/3/b83f52a0152dc39b6d966880a1d6ca824a0f3786.png)
denotes the greatest common divisor of ![$[a, b]$](http://latex.artofproblemsolving.com/d/a/2/da2e551d2ca2155b8d8f4935d2e9757722c9bab6.png)
denotes the least common multiple of 
Y by worship, Adventure10, Mango247
Let 
be a triangle with incircle 
, incenter 
and circumcircle 
. Let 
be the tangency point of with 
, let 
be the midpoint of 
, and let 
be the diametral opposite of 
with respect to . If we denote 
, then prove that the circumcircle of triangle 
is tangent to .
be a triangle with incircle 
, incenter 
and circumcircle 
. Let 
be the tangency point of with 
, let 
be the midpoint of 
, and let 
be the diametral opposite of 
with respect to . If we denote 
, then prove that the circumcircle of triangle 
is tangent to .
