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be the circumcircle of triangle 
, 
is the midpoint of 
and 
be the second intersection of 
and . Tangent line of at intersects at 
, let 
be a transversal of and 
be intersections of 
with . Let 
be a tangent line of . Prove that 
is cyclic
be positive integers that satisfy the inequality![\[
\frac{a}{b} < \frac{x}{y} < \frac{c}{d}
\]](http://latex.artofproblemsolving.com/8/d/9/8d982d52e655546ac29039fa6e440e68821bfe7f.png)
with 
. If 
, determine the minimum value of 
.
is the projection of the center of the circle on the line, then construct a square 
with two vertices 
on the line s.t. is the midpoint of 
and s.t. 
are on the same side of the line as the circle.
. They intersect the circle in some points. Take 
to be two such points s.t. 
is parallel to the line. Now let 
be the projections of 
on the line. 
is our square.
