and 
.
,![\[
\sum_{\text{cyc}} \sqrt{1 + \left(x\sqrt{1 + y^2} + y\sqrt{1 + x^2}\right)^2} \geq \sum_{\text{cyc}} xy + 2\sum_{\text{cyc}} x
\]](http://latex.artofproblemsolving.com/9/5/5/955863a0cf8747ae45b736b0631f243d3908eb84.png)
such that 
?
and 
is a subset of 
such that if 
then 
Find the smallest of 
Y by Adventure10, Mango247
In a convex quadrilateral 
prove that the nine point circles of the triangles 
are concurrent at one point.
Kostas Vittas.
PS. This result has been mentioned in the topic What angle?, by Jean-Louis.
prove that the nine point circles of the triangles 
are concurrent at one point.Kostas Vittas.
PS. This result has been mentioned in the topic What angle?, by Jean-Louis.
of 
respectively and the midpoints 
of 
.
the intersection of the 9 point circle of 
and 
,
and 
.
are concyclic (the concurrency with the other circle follows the same reasonings on angles). We will prove it by showing that 
.
.
.
and by same reasoning, 
.
