[/quote]
,
Prove that
,![\[\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}\geq 4+\frac{8(a-c)^2}{(a+b+c+d)^2}.\]](http://latex.artofproblemsolving.com/7/b/6/7b6e32d32d89ca058a06974e7276f0a5874d90a7.png)
and 
.
Let 
.
such that 
that satisfy the equation
with incenter 
.
at 
.
and 
be points on the side BC such that 
and 
,
and 
and 
,
is the Euler line of triangle 
.
,
,
,
,
are odd primes and![\[2^ap^b=(p+2)^c-1.\]](http://latex.artofproblemsolving.com/5/8/5/58518a8ebed14836b72af743ff100eae2efba5e2.png)
we have 
.
.
be reals such that 
and 
Prove that
is the reflection of 
the intersection of 
is the reflection of 
.
be the reflection of 
.
concyclic.
and 
Prove that
. Prove that: 
Y by Adventure10, Mango247
Let 
be a trapezoid with 
and 
. Let 
be the mdipoint of 
the intersection point of the diagonals of 
the intersection point of lines 
and 
be the intersection point of lines 
and 
. Prove that
i) quadrilateral
is rhombus
ii) triangles
is isosceles
iii) lines
intersects segment 
at it's midpoint
be a trapezoid with 
and 
. Let 
be the mdipoint of 
the intersection point of the diagonals of 
the intersection point of lines 
and 
be the intersection point of lines 
and 
. Prove thati) quadrilateral
is rhombusii) triangles
is isoscelesiii) lines
intersects segment 
at it's midpoint
,
then: 
is 
are midpoint of 
,
passes through midpoint of 
so 
(1)
(2) because 
-
rhombus (3)
,
so 
is Isosceles
are medians of 
(due to rhombus and b), so 
