[/quote]
,
.
and 
be the feet of the altitudes from 
to 
and from 
to 
respectively.
and 
as the reflections of 
and 
.
with 
with 
is the circumcenter of 
can be inscribed within a circle.
be a cyclic quadrilateral. Let 
be the feet of the perpendiculars from 
,
if and only if the bisectors of 
and 
are concurrent with 
be reals such that 
. Prove that
where 
are lengths of parallel sides and 
are other two sides.
and 
Prove that 
be a triangle. A circle passing through 
intersect at 
,
and 
intersect at 
.
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 
.
?
then
where 
Also, 
and 
is largest real root of the equation
then 
Equality holds iff 
and any cyclic permutations.
and 
be points on segment 
denote the circle passing through 
denote the circle passing through 
.
)
.
Y by jhu08, Adventure10, Mango247
Let 
be a convex hexagon such that 
and 
. Let 
, 
, 
, 
, 
, and 
. Suppose 
lie on the same circle 
. The circumcircles of 
and 
meet at 
and 
. The line 
meets again at 
. Prove that 
and 
are parrallel. (Here, we use 
to denote the intersection point of lines 
and 
)
be a convex hexagon such that 
and 
. Let 
, 
, 
, 
, 
, and 
. Suppose 
lie on the same circle 
. The circumcircles of 
and 
meet at 
and 
. The line 
meets again at 
. Prove that 
and 
are parrallel. (Here, we use 
to denote the intersection point of lines 
and 
)This post has been edited 1 time. Last edited by YanYau, Dec 12, 2016, 1:14 PM
and 
and 
so in quadrilateral 
:
and it results that 
and 
are similar. Also, because of the angles given 
Which means that 
.
.
so 
.
.
.
,
and 
are collinear.
and 
are cyclic. So, 
.
.
.
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By Reim's theorem 
,
as desired.
and 
are concyclic.
are collinear. This is because 
yay. So then 
by the claim, so we are done.
