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 
,
,
.
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
Let 
be the intersection point of the diagonals of the convex quadrilateral 
, 
. Points 
and 
are marked on the segments 
and 
, respectively, such that 
. Let 
and 
be the intersection points of the sides 
, 
, and the lines 
and 
respectively.
Prove that the points, , and are colinear.
be the intersection point of the diagonals of the convex quadrilateral 
, 
. Points 
and 
are marked on the segments 
and 
, respectively, such that 
. Let 
and 
be the intersection points of the sides 
, 
, and the lines 
and 
respectively.Prove that the points
, , and are colinear.
are collinear where 
that's why i proposed the possible result , but as your diagram indicates differently that 
with 
and the converse of Menelaus in the triangle 
with 
suffices to get the result
