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, 
, 
cut three common chords concurrent at 
. Consider two distinct circles 
, 
which are internally tangent to all 
. Determine, with proof, which of the following two statements is true. is the insimilicenter of and is the exsimilicenter of and .
be a parallelogram of center 
. Prove that for any point 
, there exist unique points 
and 
such that is the center of mass of 
.
with 
be given . On the extension of the segment 
beyond 
, a point 
is chosen arbitrarily. Let the line through and 
intersect the line 
at the point 
. Let 
be the intersection of the lines 
and 
. Prove that the line 
is tangent to the circumcircle of the triangle 
.
such that:
holds simultaneously.
![$f(x) = 1 + \frac{1}{x} \in (1, 2].$](http://latex.artofproblemsolving.com/b/1/5/b15c7b9c23c29917af3aba34bf47c8a5869574e8.png)
Now we suppose for a moment 
and perform casework on 
then 
so no solutions. If 
then 
so we must have 
for which we find 
If 
then 
so 
for which we find 
means 
so 
giving 
entries of 
Expanding 
of 
yields:
Or equivalently:
Note that:
Checking cases 
yields to solutions 
,