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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 
.
, 
, 
, 
, , 
, 
, 
, 
be nine points in space such that 
, 
, and 
are each regular pentagons with side length 
. Determine the lengths of the sides of triangle 
.
is equilateral; 
.
.
lies on the bisector of 
with equation 
.
and 
.
plane: 
; normal of the 
plane:
.
between both planes:![\[\cos \theta = \left|\frac{\frac{\cos 108^{o}}{\sqrt{3}}}{\sqrt{1-\frac{4\cos^{2} 108^{o}}{3}+\frac{\cos^{2} 108^{o}}{3}}}\right|=\left|\frac{1}{\sqrt{3}\tan 108^{o}}\right|=\frac{\tan 18^{o}}{\sqrt{3}}\]](http://latex.artofproblemsolving.com/d/c/3/dc33374d835500b6ba9bd5f485f8d11b254021a9.png)
of ![$\left[FC\right]: MI = \frac{1}{2} \cot 18^{o}$](http://latex.artofproblemsolving.com/2/f/3/2f34a563d0271541b805a2d9f69249a37d6712fe.png)
.
plane, 
plane: 
.
centre of the circumcircle of 
.
with radius 
equilateral with side 
.
are parallelograms because have a pair of opposite sides parallel and congruent.
.
