[/quote]
,
real numbers 
,
and transform the sequence into 
.
where 
are lengths of parallel sides and 
are other two sides.
?
be a cyclic quadrilateral. Let 
to the lines 
,
,
if and only if the bisectors of 
and 
are concurrent with 
.
be reals such that 
. Prove that
.
be the feet of the altitudes from 
to 
and 
as the reflections of 
be the circumcircle of 
.
with 
with 
is the circumcenter of 
can be inscribed within a circle.
and 
Prove that 
intersect at 
,
and 
.
if and only if 
.
and 
.
be reals such that 
Prove that
Prove that
such that 
.
of positive integers satisfying the following two conditions:
,
is in 
,
and 
;
such that 
is not divisible by 
.
Y by
In triangle 
, altitudes 
and 
are drawn, intersecting at point 
. Point 
is symmetric to 
wrt line 
. The circumscribed circles of triangles 
and 
intersect the circumscribed circle of triangle at points 
and 
, different from . The lines 
and intersect at the point 
. Prove that the lines 
and 
are parallel.
, altitudes 
and 
are drawn, intersecting at point 
. Point 
is symmetric to 
wrt line 
. The circumscribed circles of triangles 
and 
intersect the circumscribed circle of triangle at points 
and 
, different from . The lines 
and intersect at the point 
. Prove that the lines 
and 
are parallel.
,
be the midpoint of 
.
.
.
. Note that 
remains fixed , while 
and 
and 
since 
and 
are also pairs of inverses . So ,
, respectively
and 
are collinear .
and so 
is a harmonic quadrilateral , from which we get 
are tangent to 
. Hence ,![\[\measuredangle RHB=\measuredangle BCH=\measuredangle FEH\implies RH\parallel{EF}\]](http://latex.artofproblemsolving.com/5/9/8/5989ced9b0a79a49262e41fd33b2fd8e96798047.png)
. Q.E.D.
