[/quote]
,
is the midpoint of 
,
and 
is the midpoint of the segment 
.
meets the circumcircle 
.
:
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
.
.
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 
.
.
Y by rightways, Adventure10, Mango247, DEKT
Triangle 
is right angled at 
. Lines 
and 
are internal angle bisectors.
and intersect altitude 
at points 
and 
respectively.
Prove that the line which passes through the midpoints of segments
and 
is parallel to 
.
is right angled at 
. Lines 
and 
are internal angle bisectors. and intersect altitude 
at points 
and 
respectively.Prove that the line which passes through the midpoints of segments
and 
is parallel to 
.
are isosceles and, if 
are midpoints of 
and, since 
are angle bisectors, and if 
,
is midline of 
,
be midpoints of 
and 
is cyclic, so the result follows.
-
-
This way very easy.
,
and 
are cyclic. [because, 
.]
is deltoid 
.
and 
and then showed that 
from which the result follows....
,
.
,
be the midpoint of 
and 
,
be the midpoint of 
,
.
i.e. on the 
m
.
is isosceles. Therefore 
since 
is cyclic with center 
,
.
and 
are isosceles hence 
.
and 
respectively.
is 
.
.
.
and 
.
and 
.
.
.
