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.
be a triangle whose incircle is tangent to 
, 
, 
at 
, respectively. Let 
and 
be the midpoints of 
and 
, respectively. Suppose that points 
and 
lie on 
so that 
and 
. Prove that lines 
and 
intersect on the circumcircle of 
.
be an acute triangle. Points 
, and 
lie on a line in this order and satisfy 
. Let and be the midpoints of 
and 
, respectively. Suppose triangle 
is acute, and let 
be its orthocentre. Points and lie on lines 
and 
, respectively, such that 
and are concyclic and pairwise different, and 
and are concyclic and pairwise different. Prove that 
and are concyclic.
be a triangle and let every angle of be acute. Prove that 
where 
is the radius of the circumscribed circle.
we have: 
where 
and 
denote the circumradius, inradius and semiperimeter respectively. This means that in any acute-angled triangle it holds that: 
which one can rearrange as 
or 
This is clearly stronger than the proposed inequality.
tip: use 
instead of 
.
