be positive reals. Prove that there is a cyclic permutation 
of 
such that the inequality:![\[
\frac{a}{xa + yb + zc} + \frac{b}{xb + yc + za} + \frac{c}{xc + ya + zb} \geq \frac{3}{x + y + z}
\]](http://latex.artofproblemsolving.com/5/5/3/553527fc40a1b7d29dbdfb4b5e63635308f46c85.png)
and 
.
be a scalene triangle with orthocenter 
and circumcenter 
.
the midpoints of 
,
.
with diameter 
,
at a point 
.
meets line 
at 
.
and 
intersect at a point 
on 
.
Prove that: 
is the sum of 
square numbers
Let 
and 
is the sum of 
is the sum of 
Y by Adventure10, Mango247
Quadrilateral 
is inscribed in a circle. Let 
be the point of intersection of its diagonais, and 
be the midpoint of arc 
(which does not contain the other vertices of the quadrilateral). Prove that the distances from to the centers of the circles inscribed in triangles 
and 
are equal.
is inscribed in a circle. Let 
be the point of intersection of its diagonais, and 
be the midpoint of arc 
(which does not contain the other vertices of the quadrilateral). Prove that the distances from to the centers of the circles inscribed in triangles 
and 
are equal.
and 
so it suffices to show 
isosceles or simply 
,
passes through 
