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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 
.
using the Menelaus theorem to the triangle 
: 
then we take that 
using the Menelaus theorem to the triangle 
: 
then we take that 
and 
. Then clearly, 
and 
. The point is given by
Now 
has equation 
, so the point 
. Since 
and 
are collinear, we have
parallel to 
and apply thales theorem.
using the Menelaus theorem to the triangle : then we take that using the Menelaus theorem to the triangle : then we take that 
such that 
and 
.
such that 
.
.
.
