[/quote]
,
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is divided by 
.
.
intersect the sides 
and 
at the points 
and 
,
.
intersects the circumcircle of triangle 
and 
.
is twice as long as the circumradius of triangle 
such that 
is also a prime number.
be a cyclic quadrilateral. Let 
be the feet of the perpendiculars from 
to the lines 
,
,
if and only if the bisectors of 
and 
are concurrent with 
such that for all 
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\]](http://latex.artofproblemsolving.com/c/0/c/c0cdf1bcb8053f5dc26e9df6355207b5e479f861.png)
be positive integers. Prove that it is impossible to have all of the three numbers 
to be perfect squares.
is a positive real number, find the minimum of the following expression
such that for all real numbers 
,
be a quadrilateral inscribed in a circle of center 
.
and 
,
be the intersection point of the diagonals 
are distinct. Prove that the points 
are concyclic if and only if the points 
are concyclic.
such that there exists a pair 
of positive integers, such that 
is not divisible by the cube of any prime, and 
such that 
?
such that 
.
and 
on the segment 
. Consider point 
on 
such that 
. Point 
is the intersection of tangent line through point 
respectively . If the tangent through point 
intersect 
. Line 
intersect 
bisect 
.
Y by
Let triangle 
be inscribed in the circle 
. A line through point 
intersects 
and 
at points 
and 
, respectively. Let 
be the reflection of across the midpoint of , and 
be the reflection of across the midpoint of . Prove that:
a) the reflection of the orthocenter
of triangle across line 
lies on the circle .
b) the orthocenters of triangles
and 
coincide.
Im looking for a solution used complex bashing
be inscribed in the circle 
. A line through point 
intersects 
and 
at points 
and 
, respectively. Let 
be the reflection of across the midpoint of , and 
be the reflection of across the midpoint of . Prove that:a) the reflection of the orthocenter
of triangle across line 
lies on the circle .b) the orthocenters of triangles
and 
coincide.Im looking for a solution used complex bashing
,
To do part (a), we find the vectors
Then if 
,
and then
and so 
.
has equation
![$$\frac{x^2(a+c) - z(ac+x^2)}{(a-b)(ac+x^2)} = \frac{ab\left[(a+c) - \overline{z}(ac+x^2)\right]}{(a-b)(ac+x^2)}$$](http://latex.artofproblemsolving.com/a/b/7/ab72d564430466676f50fba3c89b8e21000e3c70.png)
Similarly, the altitude from 
has equation
Intersecting these gives
This point 
is the orthocenter of 
.
-- then we will have 
as well by symmetry. Now
and so
which is indeed pure imaginary. 
