[/quote]
,
:
'is a perfect square.
Prove that
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)
and 
.
Let 
.
such that 
that satisfy the equation
with incenter 
.
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)
Y by Adventure10, Mango247
Let 
be a trapezoid with 
and 
. Let 
be the midpoints of the sides 
respectively and 
be the foot of the perpendicular from point 
on 
. Prove that :
i) triangle
is isosceles
ii) midpoint
of 
is the centroid of triangle
iii) lines
and 
intersect at a point lying on line 
be a trapezoid with 
and 
. Let 
be the midpoints of the sides 
respectively and 
be the foot of the perpendicular from point 
on 
. Prove that :i) triangle
is isoscelesii) midpoint
of 
is the centroid of triangle iii) lines
and 
intersect at a point lying on line 
is origin. The lines 
give the directions of positive 
axis and 
axis respectively. 
Hence 
and 
and 
which proves part i).
The centroid of 
has coordinate 
This proves part ii).
and 
And the equation of the line joining 
These two lines intersect at the point 
Clearly, this point lies on the line joining 
(Pythagorean Theorem on 
,
)
bisects 
, 
bisects 
,
is isosceles trapezoid 
,
)
,
,
, which passes through 
.
,
which is line 
