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 three distinct points on a circle of radius 
. Prove that the triangle has an obtuse angle if and only if there exists a point 
in the plane such that the distances 
and 
are all less than .
be three distinct points on a circle of radius . Prove that the triangle has an obtuse angle if and only if there exists a point in the plane such that the distances and are all less than .
Is ,midpoint of is has obtuse angle
there exists point 
Let point satisfying 
(
) is the finding point Is acute triangle Don’t have such a point 
If is on the plane 
then 
contradiction is on the boundary. If is on 
, 
also contradiction Hence, there is no such point 
