attaining all positive values and positive numbers 
are given. Numbers 
satisfy
for every 
.
satisfying 
.
that satisfy 
with 
,
and area 
.
with diameter 
meets 
at 
,
and extend to meet 
and 
is the diameter of 
of 
,
of 
and 
Prove that
Let 
Prove that
Let 
Prove that
Let 
Prove that
.
lie on 
respectively, such that 
.
to meet the circumcircle of 
respectively. Let the circumcircles of 
meet at points 
.
to 
,
,![$[ABC]$](http://latex.artofproblemsolving.com/d/3/3/d33cc80fa8f093e155c5be46d2e5d9da3d7e1ef5.png)
.
with 
,
.
to meet the circumcircle of 
at 
,
.
Y by Adventure10, Mango247
Let 
be a quadrilateral with 
and 
. Let 
be the intersection of 
and 
. On the rays 
and 
we consider the points 
and 
respectively such that 
. We denote with 
the midpoint of the segment 
. Prove that
a)
;
b)
.
Claudiu-Stefan Popa
be a quadrilateral with 
and 
. Let 
be the intersection of 
and 
. On the rays 
and 
we consider the points 
and 
respectively such that 
. We denote with 
the midpoint of the segment 
. Prove thata)
;b)
.Claudiu-Stefan Popa
. By the way, Valentin (and also any Romanian who is either a 12th grader or already in college), I would like to know your opinion about the thread topic I am mentioning!
,
.
are perpendicular or not. Moreover, the angles 
have to be only equal, not necessarily right. From similarity of the triangles 
,
.
by the angle 
around the point 
.
become parallel and the triangles themselves centrally similar with homothety center 
.
are equal. This means that the quadrilateral 
is cyclic and the sums of its opposite angles are equal to 
.
are also similar and consequently, the quadrilateral 
is cyclic. Until now, no right angles have been used. If the diagonals 
of the cyclic quadrilateral 
is one of the maltitudes and therefore, 
.
are parallel, 
