in order to make it a perfect square?
is 
and 
.
is 
degrees, where 
and 
are the midpoints of 
and 
,
and 
.
is acute triangle. 
is orthocenter, 
,
is the midpoint of smaller arc 
is on 
such that, 
is perpendicular to 
.
is paralelogram(Synthetic sol needed).
and 
be positive real numbers such that 
.
and 
Prove that
is the midpoint of 
bisects 
and intersects 
at 
is the incenter of 
,
bisects 
and intersects the circumcircle of 
at 
,
is parallel to 
.
written on the board.At every step,Mikail chooses the two numbers 
and substitutes them with 
and 
where 
is the unchosen number on the board. Prove that at least 
negative number must be remained on the board at any step.
Y by
Two circles 
and 
with equal radii intersect at P and Q. Points B and C are located on the circles and so that they are inside the circles and , respectively. Also, points X and Y distinct from P are located on and , respectively, so that:
The intersection point of the circumcircles of triangles XPC and YPB is called S. Prove that BC, XY and QS are concurrent.
Thanks.
and 
with equal radii intersect at P and Q. Points B and C are located on the circles and so that they are inside the circles and , respectively. Also, points X and Y distinct from P are located on and , respectively, so that:
The intersection point of the circumcircles of triangles XPC and YPB is called S. Prove that BC, XY and QS are concurrent.Thanks.
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