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
, 
is the altitude from , and 
is the orthocentre. Let 
be the centre of the circle passing through and tangent to 
at . Prove that the line 
bisects .
be the second intersection of circle with . is midpoint of 
, hence done.
be the 
antipode in 
, easy to see, 
, and 
and since, 
bisects 
such that 
.
?
be the midpoint of 
intersects the perpendicular of 
,
and hence 
,
,
be midpoint of 
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= 
= 
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