be positive real numbers such that : 
. Prove that : 
,![\[
A = \left\{ r \, \middle| \, r = \frac{a}{3}\left(\frac{1}{x} + \frac{1}{y} + \frac{1}{z}\right) + b\sqrt[3]{xyz}, \quad x, y, z \in \left[1, \frac{a}{b}\right] \right\},
\]](http://latex.artofproblemsolving.com/e/9/2/e92becb6264b8e5a9753f2288436a48523a25f29.png)
and![\[
B = \left[2\sqrt{ab}, a + b\right].
\]](http://latex.artofproblemsolving.com/2/d/2/2d25aa634389b768dc26408480bb71b316c0bf17.png)
.
be an acute triangle with incentre 
and 
.
and 
intersect the circumcircle of 
and 
,
and 
such that 
and 
are parallelograms (with 
,
)
be the point of intersection of lines 
and 
.
,
Y by Adventure10, Mango247
In a convex quadrilateral 
, let 
, 
, 
, and 
be the midpoints of 
, 
, 
, and 
respectively. If 
and 
divide in four quadrilaterals with the same area, prove that is a parallelogram.
, let 
, 
, 
, and 
be the midpoints of 
, 
, 
, and 
respectively. If 
and 
divide in four quadrilaterals with the same area, prove that is a parallelogram.
is a parallelogram, ![$[MNE]=[NPE]=[PQE]=[QME]$](http://latex.artofproblemsolving.com/5/1/6/5164f5bcd2d9e330e294a1323741130e311eb05d.png)
![$[AMQ]=[BMN]=[CPN]=[DPQ],$](http://latex.artofproblemsolving.com/b/b/d/bbd0a4c3bbcc393d8c7c66382a2e97947a261c32.png)
the distances of 
to 
so 
similarly, 
is a parallelogram.
