[/quote]
,
:
be reals such that 
and 
Prove that
Where 
Prove that: 
is the sum of 
square numbers
Let 
and 
is the sum of 
is the sum of 
,![\[\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}\geq 4+\frac{8(a-c)^2}{(a+b+c+d)^2}.\]](http://latex.artofproblemsolving.com/7/b/6/7b6e32d32d89ca058a06974e7276f0a5874d90a7.png)
such that 
that satisfy the equation
with incenter 
at 
.
and 
be points on the side BC such that 
and 
,
and 
be the incenter of triangles 
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 a cyclic quadrilateral. The point 
is the reflection of 
the intersection of 
.
be the reflection of 
.
concyclic.
. Prove that: 
Y by
Let 
and 
be the circumcenter and incenter of a scalene triangle 
respectively. The incircle of touches sides 
and 
at 
and 
, respectively. Let 
and 
be the angle bisectors of so that 
and 
lie on and respectively. Let the reflections of line 
across lines 
and 
intersect at 
. Prove that 
lie on a circle.
and 
be the circumcenter and incenter of a scalene triangle 
respectively. The incircle of touches sides 
and 
at 
and 
, respectively. Let 
and 
be the angle bisectors of so that 
and 
lie on and respectively. Let the reflections of line 
across lines 
and 
intersect at 
. Prove that 
lie on a circle.
be the reflection of 
and 
b
,
belongs to the circumcircle 
(
.
is the unit circle with 
and 
and
We have 
.
![$t=\frac{(\bar{x}y-x\bar{y})(w-z)-(x-y)(\bar{w}z-w\bar{z})}{(\bar{x}-\bar{y})(w-z)-(x-y)(\bar{w}-\bar{z})}
=\frac{\bar{o}de \left[o\left(\frac{1}{d}+\frac{1}{f}\right)+(d+f)\bar{o}\right]-\bar{o}df\left[o\left(\frac{1}{d}+\frac{1}{e}\right)+(d+e)\bar{o}\right]}{|o|\left(\frac{e}{f}-\frac{f}{e} \right )}=\frac{o(de+ef+fd)+d^2ef\bar{o}}{o(e+f)}$](http://latex.artofproblemsolving.com/7/1/d/71d4460ed65db102155fbb0b2d2154514134acf7.png)
Note that 
so we have (hence 
Similarly 
and 
which is true, this completes the proof.
well known facts. The first fact is 
as it is the Euler Reflection Point of 
.
).
perpendicular bisector of 
.
is the midpoint of 
is the foot of altitude from 
.
is the rectangle. Similar to 
.
are concyclic.
be the anticomplementary triangle of 
.
is the center of 
and 
is its orthocenter and 
are the feet of altitude of the 
intersects 
again at 
and 
.
,
is perpendicular to the line joining the centers which means perpendicular to 
.
are collinear. Since 
across 
(chasing through 
. But
We can check that 
.
are concyclic.
to 
,
