[/quote]
,
,
is the midpoint of 
,
the point of intersection of the tangents at points 
and 
of the circumscribed circle of 
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 InCtrl, Adventure10, and 1 other user
Let 
be a triangle with 
. Let 
and 
be the points on the sides 
and 
, respectively, such that 
and 
. Let 
be the point in the plane such that the lines 
and 
are perpendicular to and , respectively. Prove that 
.
be a triangle with 
. Let 
and 
be the points on the sides 
and 
, respectively, such that 
and 
. Let 
be the point in the plane such that the lines 
and 
are perpendicular to and , respectively. Prove that 
.
of the orthic triangle, 
)
°
then, because 
°
and 
respectively. Let 
and 
denote the midpoints of 
and therefore it follows that 
.
in 
and 
in 
.
of 
.
,
.
.
Then again obviously 
is the orthocentre of triangle 
,
is cyclic with 
as diameter.
lies outside 
and the circumcircle of 
?
of 
so 
.
as desired.
is the contact point of the 
is the contact point of the 
and 
)
such that 
is isosceles.
,
,
angle conditions, and only 
point can simultaneously satisfy the 
denote the orthocenter of 
.
.
and similarly, 
,
.
,
is acute, and analogously for the other angles. Now, we claim that 
.
,
.
by cyclic quadrilaterals, so since 
,
,
is the circumradius of 
.
is clearly contained within the circle diameter 
,
,
or 
.
.
