[/quote]
,
.
and 
be the feet of the altitudes from 
and from 
to 
respectively.
and 
as the reflections of 
and 
be the circumcircle of 
.
with 
the intersection of ray 
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
Let 
be a convex polygon of area 
and perimeter 
. The isoperimetric ratio of is defined as 
. The roundness of is defined as 
, where 
is the maximum radius for a circle inside , and 
the minimum radius of a circle containing . Prove that 
.
be a convex polygon of area 
and perimeter 
. The isoperimetric ratio of is defined as 
. The roundness of is defined as 
, where 
is the maximum radius for a circle inside , and 
the minimum radius of a circle containing . Prove that 
.This post has been edited 1 time. Last edited by parmenides51, Oct 9, 2021, 5:48 AM
