,
,
cut three common chords concurrent at 
.
,
which are internally tangent to all 
.
and 
be positive real numbers such that 
.![\[
\frac{x^{3}}{(1 + y)(1 + z)}+\frac{y^{3}}{(1 + z)(1 + x)}+\frac{z^{3}}{(1 + x)(1 + y)}
\geq \frac{3}{4}.
\]](http://latex.artofproblemsolving.com/5/6/5/56568d27a296f1c900fabbb1042549cd2e552e29.png)
,![\[ \frac{a}{\sqrt{a^2 + 8bc}} + \frac{b}{\sqrt{b^2 + 8ca}} + \frac{c}{\sqrt{c^2 + 8ab}} \geq 1. \]](http://latex.artofproblemsolving.com/b/2/f/b2ff456632d4dd3b9ed93e4cba437bdd34d54cd8.png)
Y by Adventure10, Mango247
Four circles are drawn with the sides of quadrilateral 
as diameters. The two circles passing through 
meet again at 
. The two circles passing through 
meet again at 
. The two circles passing through 
meet again at 
. The two circles passing through 
meet again at 
. Suppose, 
are all distinct. Is the quadrilateral 
similar to ? Show with proof.
as diameters. The two circles passing through 
meet again at 
. The two circles passing through 
meet again at 
. The two circles passing through 
meet again at 
. The two circles passing through 
meet again at 
. Suppose, 
are all distinct. Is the quadrilateral 
similar to ? Show with proof.
