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 Adventure10
Circles 
are given in a plane. 
and 
touch each other externally, and both touch 
internally at 
and 
respectively. The common internal tangent to and meets at 
and 
Let 
and 
be the intersections of 
and 
with and , respectively. Prove that 
is a common tangent to 
are given in a plane. 
and 
touch each other externally, and both touch 
internally at 
and 
respectively. The common internal tangent to and meets at 
and 
Let 
and 
be the intersections of 
and 
with and , respectively. Prove that 
is a common tangent to 
This post has been edited 1 time. Last edited by IstekOlympiadTeam, Nov 28, 2015, 8:24 AM
Reason: qrong source
Reason: qrong source
be the intersection of 
parallel line 
and line 
where 
will be the intersection of 
will be the point on 
concyclic
perpendicular with parallel lines
are perpendicular lines too.
tangent with 
(We can do similar way with other side)
(Because 
(Because 
)
,
is tangent to both lines, which is a good staring exercise for the reader.
with center 
and 
and 
and 
is tangent to 
and 
is tangent to 
