AoPS Academy
. Let 
and 
be the feet of the altitudes from 
to 
and from 
to 
respectively.
and 
as the reflections of across lines 
and , respectively. Let 
be the circumcircle of 
. Denote by 
the second intersection of line 
with , and by 
the intersection of ray 
with .
is the circumcenter of , prove that , , and are collinear if and only if quadrilateral 
can be inscribed within a circle.
+1 w
of positive integers satisfying the following two conditions:
, if 
is in 
, then so are 
and 
; and contains an integer 
such that 
is not divisible by 
.
+1 w
and 
of a triangle 
intersect at a point 
. It is known that the quadrilateral 
is both inscriptable and cyclic. Prove that is an equilateral triangle.
and of a triangle intersect at a point . It is known that the quadrilateral is both inscriptable and cyclic. Prove that is an equilateral triangle.
such that 
.
?
